6.2.5 2.5

6.2.5.1 [477] problem number 1

problem number 477

Added January 2, 2019.

Problem 2.2.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a y + b x^k \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*y + b*x^k)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (b a^{-k-1} \Gamma (k+1,a x)+y e^{-a x}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := diff(w(x,y),x)+ (a*y+b*x^k)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\left (a x \right )^{-\frac {k}{2}} \left (-{\mathrm e}^{-\frac {a x}{2}} b \,x^{k} \operatorname {WhittakerM}\left (\frac {k}{2}, \frac {k}{2}+\frac {1}{2}, a x \right )+{\mathrm e}^{-a x} y a \left (a x \right )^{\frac {k}{2}} \left (k +1\right )\right )}{a \left (k +1\right )}\right )\]

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6.2.5.2 [478] problem number 2

problem number 478

Added January 2, 2019.

Problem 2.2.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a x^k y+b x^n \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*x^k*y + b*x^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
sol = Simplify[sol];
 

\[\left \{\left \{w(x,y)\to c_1\left (b (k+1)^{\frac {n-k}{k+1}} a^{-\frac {n+1}{k+1}} \Gamma \left (\frac {n+1}{k+1},\frac {a x^{k+1}}{k+1}\right )+y e^{-\frac {a x^{k+1}}{k+1}}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := diff(w(x,y),x)+ (a*x^k*y+b*x^n)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {2 \left (-\frac {{\mathrm e}^{-\frac {x \,x^{k} a}{2 k +2}} b \,x^{n} \left (k +1\right )^{2} \left (x \,x^{k} a +k +n +2\right ) \operatorname {WhittakerM}\left (\frac {-k +n}{2 k +2}, \frac {2 k +n +3}{2 k +2}, \frac {x \,x^{k} a}{k +1}\right )}{2}+\left (k +n +2\right ) \left (-\frac {\left (k +1\right ) \left (k +n +2\right ) {\mathrm e}^{-\frac {x \,x^{k} a}{2 k +2}} b \,x^{n} \operatorname {WhittakerM}\left (\frac {k +n +2}{2 k +2}, \frac {2 k +n +3}{2 k +2}, \frac {x \,x^{k} a}{k +1}\right )}{2}+{\mathrm e}^{-\frac {x \,x^{k} a}{k +1}} x^{k} \left (\frac {x \,x^{k} a}{k +1}\right )^{\frac {k +n +2}{2 k +2}} \left (n +1\right ) y a \left (k +\frac {n}{2}+\frac {3}{2}\right )\right )\right ) x^{-k} \left (\frac {x \,x^{k} a}{k +1}\right )^{\frac {-k -n -2}{2 k +2}}}{a \left (n +1\right ) \left (k +n +2\right ) \left (2 k +n +3\right )}\right )\]

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6.2.5.3 [479] problem number 3

problem number 479

Added January 2, 2019.

Problem 2.2.5.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a y^2+b x^n \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*y^2 + b*x^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol=Simplify[sol];
 

\[\left \{\left \{w(x,y)\to c_1\left (-\frac {\operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \left ((2 a x y+1) \operatorname {BesselJ}\left (\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )+\sqrt {a} \sqrt {b} x^{\frac {n}{2}+1} \left (\operatorname {BesselJ}\left (\frac {1}{n+2}-1,\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )-\operatorname {BesselJ}\left (1+\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )\right )\right )}{\operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \left ((2 a x y+1) \operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )+\sqrt {a} \sqrt {b} x^{\frac {n}{2}+1} \left (\operatorname {BesselJ}\left (-\frac {n+3}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )-\operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )\right )\right )}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := diff(w(x,y),x)+ (a*y^2+b*x^n)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-\operatorname {BesselY}\left (\frac {3+n}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right ) \sqrt {a b}\, x^{\frac {n}{2}+1}+\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right ) y a x +\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right )}{\operatorname {BesselJ}\left (\frac {3+n}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right ) \sqrt {a b}\, x^{\frac {n}{2}+1}-\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right ) y a x -\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right )}\right )\]

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6.2.5.4 [480] problem number 4

problem number 480

Added January 2, 2019.

Problem 2.2.5.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( y^2+a n x^{n-1} -a^2 x^{2 n} \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + a*n*x^(n - 1) - a^2*x^(2*n))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {(-2)^{\frac {1}{n+1}} (n+1) a^{\frac {1}{n+1}} \left (a x^n-y\right )}{(-2)^{\frac {1}{n+1}} (n+1) a^{\frac {1}{n+1}} e^{\frac {2 a x^{n+1}}{n+1}}+(n+1)^{\frac {1}{n+1}} \left (a x^n-y\right ) \Gamma \left (\frac {1}{n+1},-\frac {2 a x^{n+1}}{n+1}\right )}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := diff(w(x,y),x)+ (y^2+a*n*x^(n-1)-a^2*x^(2*n))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (-\frac {2 x^{\frac {3 n}{2}+2} \left (x^{n} a -y \right )}{-3 \left (n +2\right ) {\mathrm e}^{\frac {a \,x^{n +1}}{n +1}} \left (a \left (n +\frac {4}{3}\right ) x^{n +1}+\frac {\left (n +2\right ) \left (-y x +n +1\right )}{3}\right ) \operatorname {WhittakerM}\left (\frac {n +2}{2 n +2}, \frac {3+2 n}{2 n +2}, -\frac {2 a \,x^{n +1}}{n +1}\right )+2 \left (a^{2} x^{2 n +2}-\frac {a \left (n +2\right ) x^{n +1}}{2}-a y \,x^{n +2}-\frac {\left (n +2\right ) \left (-y x +n +1\right )}{2}\right ) \left (n +1\right ) {\mathrm e}^{\frac {a \,x^{n +1}}{n +1}} \operatorname {WhittakerM}\left (-\frac {n}{2 n +2}, \frac {3+2 n}{2 n +2}, -\frac {2 a \,x^{n +1}}{n +1}\right )+2 \left (n +2\right )^{2} \left (n +\frac {3}{2}\right ) \left (-\frac {2 a \,x^{n +1}}{n +1}\right )^{\frac {4+3 n}{2 n +2}} {\mathrm e}^{\frac {2 a \,x^{n +1}}{n +1}}}\right )\]

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6.2.5.5 [481] problem number 5

problem number 481

Added January 2, 2019.

Problem 2.2.5.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( y^2 + a x^n y + a x^{n-1} \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + a*x^n*y + a*x^(n - 1))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
sol = Simplify[sol];
 

Failed

Maple ✓

restart; 
pde := diff(w(x,y),x)+ (y^2+a*x^n*y+a*x^(n-1))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (-\frac {{\mathrm e}^{\frac {a x \,x^{n}}{2 n +2}} \left (\left (n +1\right )^{2} \left (x y +1\right ) \left (a x \,x^{n}-n \right ) \operatorname {WhittakerM}\left (\frac {-n -2}{2 n +2}, \frac {2 n +1}{2 n +2}, -\frac {a x \,x^{n}}{n +1}\right )-2 \left (\frac {\left (x y +1\right ) \left (n +1\right ) n \operatorname {WhittakerM}\left (\frac {n}{2 n +2}, \frac {2 n +1}{2 n +2}, -\frac {a x \,x^{n}}{n +1}\right )}{2}+a x \,x^{n} {\mathrm e}^{\frac {a x \,x^{n}}{2 n +2}} \left (-\frac {a x \,x^{n}}{n +1}\right )^{\frac {n}{2 n +2}} \left (n +\frac {1}{2}\right )\right ) n \right ) x^{-n} \left (-\frac {a x \,x^{n}}{n +1}\right )^{-\frac {n}{2 n +2}}}{a n \,x^{2} \left (x y +1\right ) \left (2 n +1\right )}\right )\]

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6.2.5.6 [482] problem number 6

problem number 482

Added January 2, 2019.

Problem 2.2.5.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( y^2+a x^n y-a b x^n -b^2 \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + a*x^n*y - a*b*x^n - b^2)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := diff(w(x,y),x)+ (y^2+a*x^n*y-a*b*x^n-b^2)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\left (-b +y \right ) \int {\mathrm e}^{\frac {x \left (2 b \left (n +1\right )+x^{n} a \right )}{n +1}}d x +{\mathrm e}^{\frac {x \left (2 b \left (n +1\right )+x^{n} a \right )}{n +1}}}{b -y}\right )\]

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6.2.5.7 [483] problem number 7

problem number 483

Added January 2, 2019.

Problem 2.2.5.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a x^n y^2 + b x^{-n-2} \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*x^n*y^2 + b*x^(-n - 2))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {x^{\sqrt {(n+1)^2-4 a b}} \left (\sqrt {(n+1)^2-4 a b}+2 a y x^{n+1}+n+1\right )}{\sqrt {(n+1)^2-4 a b}-2 a y x^{n+1}-n-1}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := diff(w(x,y),x)+ (a*x^n*y^2+b*x^(-n-2))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\ln \left (x \right ) \sqrt {4 b a -n^{2}-2 n -1}-2 \arctan \left (\frac {2 a y \,x^{n +1}+n +1}{\sqrt {4 b a -n^{2}-2 n -1}}\right )}{\sqrt {4 b a -n^{2}-2 n -1}}\right )\]

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6.2.5.8 [484] problem number 8

problem number 484

Added January 2, 2019.

Problem 2.2.5.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a x^n y^2 + b m x^{m-1} -a b^2 x^{n+2 m} \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*x^n*y^2 + b*m*x^(m - 1) - a*b^2*x^(n + 2*m))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {(-2)^{\frac {n+1}{m+n+1}} (m+n+1) a^{\frac {m+2 n+2}{m+n+1}} b^{\frac {n+1}{m+n+1}} \left (b x^m-y\right )}{(-2)^{\frac {n+1}{m+n+1}} (m+n+1) a^{\frac {n+1}{m+n+1}} b^{\frac {n+1}{m+n+1}} e^{\frac {2 a b x^{m+n+1}}{m+n+1}}+a (m+n+1)^{\frac {n+1}{m+n+1}} \left (b x^m-y\right ) \Gamma \left (\frac {n+1}{m+n+1},-\frac {2 a b x^{m+n+1}}{m+n+1}\right )}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := diff(w(x,y),x)+ (a*x^n*y^2 + b*m*x^(m-1) -a*b^2*x^(n+2*m))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {2 a \,x^{\frac {3 m}{2}+2 n +2} \left (x^{m} b -y \right )}{3 \left (a b \left (m +\frac {4 n}{3}+\frac {4}{3}\right ) x^{m +n +1}-\frac {\left (m +2 n +2\right ) \left (a y \,x^{1+n}-m -n -1\right )}{3}\right ) \left (m +2 n +2\right ) {\mathrm e}^{\frac {a b \,x^{m +n +1}}{m +n +1}} \operatorname {WhittakerM}\left (\frac {m +2 n +2}{2 n +2 m +2}, \frac {2 m +3 n +3}{2 n +2 m +2}, -\frac {2 a b \,x^{m +n +1}}{m +n +1}\right )+2 \left (-a^{2} x^{2 n +2 m +2} b^{2}+a^{2} x^{m +2 n +2} b y +\frac {\left (m +2 n +2\right ) \left (a b \,x^{m +n +1}-a y \,x^{1+n}+m +n +1\right )}{2}\right ) \left (m +n +1\right ) {\mathrm e}^{\frac {a b \,x^{m +n +1}}{m +n +1}} \operatorname {WhittakerM}\left (-\frac {m}{2 n +2 m +2}, \frac {2 m +3 n +3}{2 n +2 m +2}, -\frac {2 a b \,x^{m +n +1}}{m +n +1}\right )-2 \,{\mathrm e}^{\frac {2 a b \,x^{m +n +1}}{m +n +1}} \left (m +\frac {3 n}{2}+\frac {3}{2}\right ) \left (m +2 n +2\right )^{2} \left (-\frac {2 a b \,x^{m +n +1}}{m +n +1}\right )^{\frac {3 m +4 n +4}{2 n +2 m +2}}}\right )\]

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6.2.5.9 [485] problem number 9

problem number 485

Added January 2, 2019.

Problem 2.2.5.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( (n+1)x^n y^2 - a x^{n+m+1} y + a x^m \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + ((n + 1)*x^n*y^2 - a*x^(n + m + 1)*y + a*x^m)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := diff(w(x,y),x)+ ((n+1)*x^n*y^2 - a*x^(n+m+1)* y + a*x^m)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (-\frac {\left (a x \,x^{m} \left (-n +m \right ) \operatorname {hypergeom}\left (\left [\frac {2+2 m}{m +n +2}\right ], \left [\frac {3+2 m +n}{m +n +2}\right ], \frac {a \,x^{m} x^{n} x^{2}}{m +n +2}\right )-\left (1+m \right ) \operatorname {hypergeom}\left (\left [\frac {-n +m}{m +n +2}\right ], \left [\frac {1+m}{m +n +2}\right ], \frac {a \,x^{m} x^{n} x^{2}}{m +n +2}\right ) \left (a \,x^{m} x -y \left (n +1\right )\right )\right ) \left (m +2 n +3\right )}{\left (1+m \right ) \left (a \,x^{2} x^{m} x^{n} \left (1+m \right ) \operatorname {hypergeom}\left (\left [\frac {3+2 m +n}{m +n +2}\right ], \left [\frac {2 m +3 n +5}{m +n +2}\right ], \frac {a \,x^{m} x^{n} x^{2}}{m +n +2}\right )-\operatorname {hypergeom}\left (\left [\frac {1+m}{m +n +2}\right ], \left [\frac {m +2 n +3}{m +n +2}\right ], \frac {a \,x^{m} x^{n} x^{2}}{m +n +2}\right ) \left (x \left (a \,x^{m} x -y \left (n +1\right )\right ) x^{n}-n -1\right ) \left (m +2 n +3\right )\right )}\right )\]

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6.2.5.10 [486] problem number 10

problem number 486

Added January 2, 2019.

Problem 2.2.5.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a x^n y^2 + b x^m y+ b c x^m -a c^2 x^n \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*x^n*y^2 + b*x^m*y + b*c*x^m - a*c^2*x^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := diff(w(x,y),x)+ (a*x^n*y^2 + b*x^m*y+ b*c*x^m -a*c^2*x^n)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-\int x^{n} {\mathrm e}^{-\frac {2 x \left (-\frac {b \left (n +1\right ) x^{m}}{2}+x^{n} a c \left (m +1\right )\right )}{\left (m +1\right ) \left (n +1\right )}}d x a \left (c +y \right )-{\mathrm e}^{-\frac {2 x \left (-\frac {b \left (n +1\right ) x^{m}}{2}+x^{n} a c \left (m +1\right )\right )}{\left (m +1\right ) \left (n +1\right )}}}{c +y}\right )\]

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6.2.5.11 [487] problem number 11

problem number 487

Added January 2, 2019.

Problem 2.2.5.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a x^n y^2-a x^n (b x^m +c) y+ b m x^{m-1} \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*x^n*y^2 - a*x^n*(b*x^m + c)*y + b*m*x^(m - 1))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (-\frac {b m \left (b x^m+c\right ) \left (b x^m+c-y\right )}{b m \left (b x^m+c\right ) \left (b x^m+c-y\right ) \int _1^x\frac {\exp \left (a K[1]^{n+1} \left (\frac {b K[1]^m}{m+n+1}+\frac {c}{n+1}\right )\right ) K[1]^{m-1}}{\left (b K[1]^m+c\right )^2}dK[1]-y e^{a x^{n+1} \left (\frac {b x^m}{m+n+1}+\frac {c}{n+1}\right )}}\right )\right \}\right \}\]

Maple ✗

restart; 
pde := diff(w(x,y),x)+ (a*x^n*y^2-a*x^n*(b*x^m +c)*y+ b*m*x^(m-1))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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6.2.5.12 [488] problem number 12

problem number 488

Added January 2, 2019.

Problem 2.2.5.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x - \left (a n x^{n-1} y^2 - c x^m (a x^n+b) + c x^m \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] - (a*n*x^(n - 1)*y^2 - c*x^m*(a*x^n + b) + c*x^m)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde := diff(w(x,y),x)- (a*n*x^(n-1)*y^2 - c*x^m*(a*x^n+b) + c*x^m)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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6.2.5.13 [489] problem number 13

problem number 489

Added January 2, 2019.

Problem 2.2.5.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a x^n y^2+b x^m y+ c k x^{k-1}-b c x^{m+k}-a c^2 x^{n+2 k} \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*x^n*y^2 + b*x^m*y + c*k*x^(k - 1) - b*c*x^(m + k) - a*c^2*x^(n + 2*k))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde := diff(w(x,y),x)+ (a*x^n*y^2+b*x^m*y+ c*k*x^(k-1)-b*c*x^(m+k)-a*c^2*x^(n+2*k))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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6.2.5.14 [490] problem number 14

problem number 490

Added January 2, 2019.

Problem 2.2.5.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a x^{2 n+1} y^3 + b x^{-n-2} \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*x^(2*n + 1)*y^3 + b*x^(-n - 2))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := diff(w(x,y),x)+ (a*x^(2*n+1)*y^3 + b*x^(-n-2))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\ln \left (x \right )-\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{3}+\left (n +1\right ) \textit {\_Z} +b \right )}{\sum }\frac {\ln \left (y \,x^{n +1}-\textit {\_R} \right )}{3 \textit {\_R}^{2} a +n +1}\right )\right )\]

Solution contains RootOf

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6.2.5.15 [491] problem number 15

problem number 491

Added January 2, 2019.

Problem 2.2.5.15 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a x^n y^3 + 3 a b x^{n+m} y^2 - b m x^{m-1} - 2 a b^3 x^{n+3 m} \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*x^n*y^3 + 3*a*b*x^(n + m)*y^2 - b*m*x^(m - 1) - 2*a*b^3*x^(n + 3*m))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {6^{-\frac {n+1}{2 m+n+1}} (2 m+n+1)^{-\frac {2 m}{2 m+n+1}} b^{-\frac {2 (n+1)}{2 m+n+1}} e^{-\frac {6 a b^2 x^{2 m+n+1}}{2 m+n+1}} \left (6^{\frac {n+1}{2 m+n+1}} (2 m+n+1)^{\frac {2 m}{2 m+n+1}} b^{\frac {2 (n+1)}{2 m+n+1}}-2 a^{\frac {2 m}{2 m+n+1}} \left (b x^m+y\right )^2 e^{\frac {6 a b^2 x^{2 m+n+1}}{2 m+n+1}} \Gamma \left (\frac {n+1}{2 m+n+1},\frac {6 a b^2 x^{2 m+n+1}}{2 m+n+1}\right )\right )}{\left (b x^m+y\right )^2}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := diff(w(x,y),x)+ (a*x^n*y^3 + 3*a*b*x^(n+m)*y^2 - b*m*x^(m-1) - 2*a*b^3*x^(n+3*m))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {24 \,2^{\frac {-m -n -1}{2 m +n +1}} \left (\frac {a \,x^{2 m +n +1} b^{2}}{2 m +n +1}\right )^{\frac {-m -n -1}{2 m +n +1}} x^{-2 m -n -1} 3^{\frac {-3 m -2 n -2}{2 m +n +1}} \left (\frac {\left (m +n +1\right )^{2} \left (m +\frac {n}{2}+\frac {1}{2}\right ) \left (\frac {x^{2 m +n +1} b^{2}}{2}+y \left (x^{m +n +1} b +\frac {x^{n +1} y}{2}\right )\right ) \operatorname {WhittakerM}\left (\frac {m +n +1}{2 m +n +1}, \frac {4 m +3 n +3}{4 m +2 n +2}, \frac {6 a \,x^{2 m +n +1} b^{2}}{2 m +n +1}\right )}{3}+\left (m +\frac {n}{2}+\frac {1}{2}\right )^{2} \left (x^{3 m +2 n +2} y a \,b^{3}+\frac {x^{4 m +2 n +2} a \,b^{4}}{2}+\frac {x^{2 n +2+2 m} y^{2} a \,b^{2}}{2}+\frac {\left (m +n +1\right ) \left (\frac {x^{2 m +n +1} b^{2}}{2}+y \left (x^{m +n +1} b +\frac {x^{n +1} y}{2}\right )\right )}{3}\right ) \operatorname {WhittakerM}\left (-\frac {m}{2 m +n +1}, \frac {4 m +3 n +3}{4 m +2 n +2}, \frac {6 a \,x^{2 m +n +1} b^{2}}{2 m +n +1}\right )+\frac {\left (n +1\right ) 2^{\frac {m +n +1}{2 m +n +1}} \left (m +n +1\right ) x^{2 m +n +1} \left (\frac {a \,x^{2 m +n +1} b^{2}}{2 m +n +1}\right )^{\frac {m +n +1}{2 m +n +1}} \left (m +\frac {3 n}{4}+\frac {3}{4}\right ) 3^{\frac {m +n +1}{2 m +n +1}} {\mathrm e}^{-\frac {3 a \,x^{2 m +n +1} b^{2}}{2 m +n +1}} b^{2}}{2}\right ) {\mathrm e}^{-\frac {3 a \,x^{2 m +n +1} b^{2}}{2 m +n +1}}}{b^{2} \left (n +1\right ) \left (m +n +1\right ) \left (4 m +3 n +3\right ) \left (x^{2 m} b^{2}+2 y \,x^{m} b +y^{2}\right )}\right )\]

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6.2.5.16 [492] problem number 16

problem number 492

Added January 2, 2019.

Problem 2.2.5.16 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a x^n y^3 + 3 a b x^{n+m} y^2+ c x^k y- 2 a b^3 x^{n+3 m} + b c x^{m+l} - b m x^{m-1} \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*x^n*y^3 + 3*a*b*x^(n + m)*y^2 + c*x^k*y - 2*a*b^3*x^(n + 3*m) + b*c*x^(m + l) - b*m*x^(m - 1))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde := diff(w(x,y),x)+ (a*x^n*y^3 + 3*a*b*x^(n+m)*y^2+ c*x^k*y-2*a*b^3*x^(n+3*m) + b*c*x^(m+l)-b*m*x^(m-1))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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6.2.5.17 [493] problem number 17

problem number 493

Added January 2, 2019.

Problem 2.2.5.17 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a y^n + b x ^{\frac {n}{1-n}} \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*y^n + b*x^(n/(1 - n)))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := diff(w(x,y),x)+ (a*y^n+b*x^(n/(1-n)))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (x^{\frac {n}{-1+n}} \int _{\textit {\_b}}^{y}\frac {1}{a \,\textit {\_a}^{n} \left (-1+n \right ) x^{\frac {-1+2 n}{-1+n}}+x^{\frac {n}{-1+n}} \textit {\_a} +b x \left (-1+n \right )}d \textit {\_a} \right )\]

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6.2.5.18 [494] problem number 18

problem number 494

Added January 2, 2019.

Problem 2.2.5.18 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a x^{m-n-(m n)} y^n + b x^m \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*x^(m - n - m*n)*y^n + b*x^m)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := diff(w(x,y),x)+ (a*x^(m-n-(m*n))*y^n + b*x^m)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (-x^{n \left (m +1\right )} \int _{\textit {\_b}}^{y}\frac {1}{b \,x^{\left (n +1\right ) \left (m +1\right )}-\textit {\_a} \left (m +1\right ) x^{n \left (m +1\right )}+a \,x^{m +1} \textit {\_a}^{n}}d \textit {\_a} +\ln \left (x \right )\right )\]

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6.2.5.19 [495] problem number 19

problem number 495

Added January 2, 2019.

Problem 2.2.5.19 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( a x^n y^k + b x^m y \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*x^n*y^k + b*x^m*y)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y^{1-k} e^{\frac {b (k-1) x^{m+1}}{m+1}}-a (-1)^{-\frac {n+1}{m+1}} (m+1)^{\frac {n-m}{m+1}} b^{-\frac {n+1}{m+1}} (k-1)^{\frac {m-n}{m+1}} \Gamma \left (\frac {n+1}{m+1},-\frac {b (k-1) x^{m+1}}{m+1}\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde := diff(w(x,y),x)+ (a*x^n*y^k + b*x^m*y)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\left (-\frac {x^{m} x b \left (k -1\right )}{m +1}\right )^{\frac {-m -n -2}{2 m +2}} x^{-m} \left (\left (m +1\right )^{2} \left (x^{m} x b \left (k -1\right )-m -n -2\right ) x^{n} {\mathrm e}^{\frac {x^{m} x b \left (k -1\right )}{2 m +2}} a \operatorname {WhittakerM}\left (\frac {-m +n}{2 m +2}, \frac {2 m +n +3}{2 m +2}, -\frac {x^{m} x b \left (k -1\right )}{m +1}\right )-\left (m +n +2\right ) \left ({\mathrm e}^{\frac {x^{m} x b \left (k -1\right )}{2 m +2}} a \,x^{n} \left (m +1\right ) \left (m +n +2\right ) \operatorname {WhittakerM}\left (\frac {m +n +2}{2 m +2}, \frac {2 m +n +3}{2 m +2}, -\frac {x^{m} x b \left (k -1\right )}{m +1}\right )-2 \,{\mathrm e}^{\frac {x^{m} x b \left (k -1\right )}{m +1}} \left (m +\frac {n}{2}+\frac {3}{2}\right ) b \,x^{m} \left (-\frac {x^{m} x b \left (k -1\right )}{m +1}\right )^{\frac {m +n +2}{2 m +2}} y \,y^{-k} \left (n +1\right )\right )\right )}{b \left (n +1\right ) \left (m +n +2\right ) \left (2 m +n +3\right )}\right )\]

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6.2.5.20 [496] problem number 20

problem number 496

Added January 2, 2019.

Problem 2.2.5.20 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( a y^2 + b y+ c x^{2 b} \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + (a*y^2 + b*y + c*x^(2*b))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {\sqrt {a} y \sin \left (\frac {\sqrt {a} \sqrt {c} x^b}{b}\right )+\sqrt {c} x^b \cos \left (\frac {\sqrt {a} \sqrt {c} x^b}{b}\right )}{\sqrt {c} x^b \sin \left (\frac {\sqrt {a} \sqrt {c} x^b}{b}\right )-\sqrt {a} y \cos \left (\frac {\sqrt {a} \sqrt {c} x^b}{b}\right )}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := x*diff(w(x,y),x)+ (a*y^2 + b*y+ c*x^(2*b))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {x^{b} \sqrt {a}\, \sqrt {c}-\arctan \left (\frac {x^{-b} \sqrt {a}\, y}{\sqrt {c}}\right ) b}{b}\right )\]

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6.2.5.21 [497] problem number 21

problem number 497

Added January 2, 2019.

Problem 2.2.5.21 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( a y^2+(n+b x^n) y + c x^{2 n} \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + (a*y^2 + (n + b*x^n)*y + c*x^(2*n))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {e^{\frac {x^n \sqrt {b^2-4 a c}}{n}} \left (x^n \sqrt {b^2-4 a c}+2 a y+b x^n\right )}{x^n \sqrt {b^2-4 a c}-2 a y-b x^n}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := x*diff(w(x,y),x)+ (a*y^2+(n+b*x^n)*y + c*x^(2*n))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (-\frac {b \left (-2 b n \arctan \left (\frac {b^{2}+2 x^{-n} y a b}{\sqrt {4 b^{2} a c -b^{4}}}\right )+x^{n} \sqrt {4 b^{2} a c -b^{4}}\right )}{\sqrt {4 b^{2} a c -b^{4}}\, n}\right )\]

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6.2.5.22 [498] problem number 22

problem number 498

Added January 2, 2019.

Problem 2.2.5.22 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( a x^n y^2 + b y+ c x^{-n} \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + (a*x^n*y^2 + b*y + c/x^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (-\frac {x^{\sqrt {-4 a c+b^2+2 b n+n^2}} \left (\sqrt {-4 a c+b^2+2 b n+n^2}+2 a y x^n+b+n\right )}{-\sqrt {-4 a c+b^2+2 b n+n^2}+2 a y x^n+b+n}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := x*diff(w(x,y),x)+ (a*x^n*y^2+b*y+c*x^(-n))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\ln \left (x \right ) \sqrt {4 c a -b^{2}-2 b n -n^{2}}-2 \arctan \left (\frac {2 a y \,x^{n}+b +n}{\sqrt {4 c a -b^{2}-2 b n -n^{2}}}\right )}{\sqrt {4 c a -b^{2}-2 b n -n^{2}}}\right )\]

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6.2.5.23 [499] problem number 23

problem number 499

Added January 2, 2019.

Problem 2.2.5.23 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( a x^n y^2+ m y- a b^2 x^{x+2 m} \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + (a*x^n*y^2 + m*y - a*b^2*x^(x + 2*m))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := x*diff(w(x,y),x)+  (a*x^n*y^2+ m*y- a*b^2*x^(x+2*m))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\left (-2 a \,x^{n} y -m -n \right ) \operatorname {BesselI}\left (\frac {-m -n}{n +x +2 m}, \frac {2 a b \,x^{\frac {n}{2}+\frac {x}{2}+m}}{n +x +2 m}\right )-2 x \left (\frac {d}{d x}\operatorname {BesselI}\left (\frac {-m -n}{n +x +2 m}, \frac {2 a b \,x^{\frac {n}{2}+\frac {x}{2}+m}}{n +x +2 m}\right )\right )}{\left (2 a \,x^{n} y +m +n \right ) \operatorname {BesselK}\left (\frac {m +n}{n +x +2 m}, \frac {2 a b \,x^{\frac {n}{2}+\frac {x}{2}+m}}{n +x +2 m}\right )+2 x \left (\frac {d}{d x}\operatorname {BesselK}\left (\frac {m +n}{n +x +2 m}, \frac {2 a b \,x^{\frac {n}{2}+\frac {x}{2}+m}}{n +x +2 m}\right )\right )}\right )\]

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6.2.5.24 [500] problem number 24

problem number 500

Added January 2, 2019.

Problem 2.2.5.24 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( x^{2 n} y^2+(m-n) y+ x^{2 m} \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + (x^(2*n)*y^2 + (m - n)*y + x^(2*m))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\arctan \left (y x^{n-m}\right )-\frac {x^{m+n}}{m+n}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := x*diff(w(x,y),x)+  (x^(2*n)*y^2+(m-n)*y+ x^(2*m))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\left (-m -n \right ) \arctan \left (x^{n -m} y \right )+x^{m +n}}{m +n}\right )\]

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6.2.5.25 [501] problem number 25

problem number 501

Added January 2, 2019.

Problem 2.2.5.25 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( a x^{2 n} y^2+ (b x^n -n) y + c \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + (a*x^(2*n)*y^2 + (b*x^n - n)*y + c)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {e^{\frac {x^n \sqrt {b^2-4 a c}}{n}} \left (\sqrt {b^2-4 a c}+2 a y x^n+b\right )}{\sqrt {b^2-4 a c}-2 a y x^n-b}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := x*diff(w(x,y),x)+ (a*x^(2*n)*y^2+ (b*x^n -n)*y + c)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {2 b \left (\arctan \left (\frac {b^{2}+2 x^{n} y a b}{\sqrt {4 b^{2} a c -b^{4}}}\right ) b n -\frac {x^{n} \sqrt {4 b^{2} a c -b^{4}}}{2}\right )}{\sqrt {4 b^{2} a c -b^{4}}\, n}\right )\]

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6.2.5.26 [502] problem number 26

problem number 502

Added January 2, 2019.

Problem 2.2.5.26 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( a x^{2 n + m} y^2 +(b x^{n+m}-n) y+ c x^m \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + (a*x^(2*n + m)*y^2 + (b*x^(n + m) - n)*y + c*x^m)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {e^{\frac {\sqrt {b^2-4 a c} x^{m+n}}{m+n}} \left (\sqrt {b^2-4 a c}+2 a y x^n+b\right )}{\sqrt {b^2-4 a c}-2 a y x^n-b}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := x*diff(w(x,y),x)+ (a*x^(2*n + m)*y^2 +(b*x^(n+m)-n)*y+ c*x^m)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (-\frac {\left (-2 b \left (n +m \right ) \arctan \left (\frac {b^{2}+2 y \,x^{n} a b}{\sqrt {4 b^{2} a c -b^{4}}}\right )+x^{n +m} \sqrt {4 b^{2} a c -b^{4}}\right ) b}{\sqrt {4 b^{2} a c -b^{4}}\, \left (n +m \right )}\right )\]

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6.2.5.27 [503] problem number 27

problem number 503

Added January 2, 2019.

Problem 2.2.5.27 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( a y^3+3 a b x^n y^2 - b n x^n -2 a b^3 x^{3 n} \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + (a*y^3 + 3*a*b*x^n*y^2 - b*n*x^n - 2*a*b^3*x^(3*n))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {e^{-\frac {3 a b^2 x^{2 n}}{n}} \left (a e^{\frac {3 a b^2 x^{2 n}}{n}} \left (b x^n+y\right )^2 \operatorname {ExpIntegralEi}\left (-\frac {3 a b^2 x^{2 n}}{n}\right )+n\right )}{n \left (b x^n+y\right )^2}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := x*diff(w(x,y),x)+ (a*y^3+3*a*b*x^n*y^2 - b*n*x^n -2*a*b^3*x^(3*n) )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-\operatorname {Ei}_{1}\left (\frac {3 b^{2} a \,x^{2 n}}{n}\right ) a \left (x^{2 n} b^{2}+2 y \,x^{n} b +y^{2}\right )+{\mathrm e}^{-\frac {3 b^{2} a \,x^{2 n}}{n}} n}{n \left (x^{2 n} b^{2}+2 y \,x^{n} b +y^{2}\right )}\right )\]

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6.2.5.28 [504] problem number 28

problem number 504

Added January 2, 2019.

Problem 2.2.5.28 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( a x^{2 n +1} y^3 + (b x -n) y + c x^{1-n} \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + (a*x^(2*n + 1)*y^3 + (b*x - n)*y + c*x^(1 - n))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := x*diff(w(x,y),x)+ (a*x^(2*n +1)*y^3 + (b*x-n)*y + c*x^(1-n) )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (b \left (\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,c^{2} \textit {\_Z}^{3}+b^{3} \textit {\_Z} -b^{3}\right )}{\sum }\frac {\ln \left (\frac {-y \,x^{n} b -\textit {\_R} c}{c}\right )}{3 \textit {\_R}^{2} a \,c^{2}+b^{3}}\right ) b^{2}-x \right )\right )\]

Solution contains RootOf

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6.2.5.29 [505] problem number 29

problem number 505

Added January 2, 2019.

Problem 2.2.5.29 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( a x^{n+2} y^3+ (b x^n-1) y + c x^{n-1} \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + (a*x^(n + 2)*y^3 + (b*x^n - 1)*y + c*x^(n - 1))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := x*diff(w(x,y),x)+ (a*x^(n+2)*y^3+ (b*x^n-1)*y + c*x^(n-1) )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {b \left (\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} a \,c^{2}+\textit {\_Z} \,b^{3}-b^{3}\right )}{\sum }\frac {\ln \left (\frac {-y b x -\textit {\_R} c}{c}\right )}{3 \textit {\_R}^{2} a \,c^{2}+b^{3}}\right ) b^{2} n -x^{n}\right )}{n}\right )\]

Solution contains RootOf

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6.2.5.30 [506] problem number 30

problem number 506

Added January 2, 2019.

Problem 2.2.5.30 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( y+a x^{n - m }y^m+b x^{n-k} y^k \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + (y + a*x^(n - m)*y^m + b*x^(n - k)*y^k)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := x*diff(w(x,y),x)+ ( y+a*x^(n - m)*y^m+b*x^(n-k)*y^k )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-x^{m +k} \left (n -1\right ) \int _{\textit {\_b}}^{y}\frac {1}{\textit {\_a}^{m} x^{k} a +x^{m} \textit {\_a}^{k} b}d \textit {\_a} +x^{n}}{x \left (n -1\right )}\right )\]

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6.2.5.31 [507] problem number 31

problem number 507

Added January 2, 2019.

Problem 2.2.5.31 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ y w_x + \left (x^{n-1}((1+2 n)x+a n) y-n x^{2 n}(x+a) \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  y*D[w[x, y], x] + (x^(n - 1)*((1 + 2*n)*x + a*n)*y - n*x^(2*n)*(x + a))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := y*diff(w(x,y),x)+ ( x^(n-1)*((1+2*n)*x+a*n)*y-n*x^(2*n)*(x+a) )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {n \left (a +\frac {x}{2}\right ) {\mathrm e}^{\frac {2 \arctan \left (\frac {n \left (x^{n} \left (2 a +x \right )-y \right )}{\sqrt {-n^{2}}\, \left (x^{n} x -y \right )}\right )}{\sqrt {-n^{2}}}}+{\mathrm e}^{-\textit {\_a}} \tan \left (\frac {\textit {\_a} \sqrt {-n^{2}}}{2}\right ) \arctan \left (\frac {n \left (x^{n} \left (2 a +x \right )-y \right )}{\sqrt {-n^{2}}\, \left (x^{n} x -y \right )}\right ) x}{x}\right )\]

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6.2.5.32 [508] problem number 32

problem number 508

Added January 2, 2019.

Problem 2.2.5.32 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ y w_x + \left ( (a(2 n +k)x^k+b)x^{n-1}y -(a^2 n x^{2 k}+ a b x^k -c) x^{2 n-1} \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  y*D[w[x, y], x] + ((a*(2*n + k)*x^k + b)*x^(n - 1)*y - (a^2*n*x^(2*k) + a*b*x^k - c)*x^(2*n - 1))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde := y*diff(w(x,y),x)+ ( (a*(2*n+k)*x^k+b)*x^(n-1)*y -(a^2*n*x^(2*k)+ a*b*x^k-c)*x^(2*n-1) )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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6.2.5.33 [509] problem number 33

problem number 509

Added January 2, 2019.

Problem 2.2.5.33 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x(2 a x y+b) w_x - \left ( a(m+3) x y^2+b(m+2)y-c x^m \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x*(2*a*x*y + b)*D[w[x, y], x] - (a*(m + 3)*x*y^2 + b*(m + 2)*y - c*x^m)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {x^{m+2} \left (2 (m+1) y (a x y+b)-c x^m\right )}{2 a (m+1)}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := x*(2*a*x*y+b)*diff(w(x,y),x)- ( a*(m+3)*x*y^2+b*(m+2)*y-c*x^m )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (-\frac {2 x^{2} x^{m} \left (-\frac {c \,x^{m}}{2}+y \left (m +1\right ) \left (a x y +b \right )\right )}{2 m +2}\right )\]

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6.2.5.34 [510] problem number 34

problem number 510

Added January 2, 2019.

Problem 2.2.5.34 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x^2(2 a x y+b) w_x - \left ( 4 a x^2 y^2 + 3 b x y-c x^2 - k \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x^2*(2*a*x*y + b)*D[w[x, y], x] - (4*a*x^2*y^2 + 3*b*x*y - c*x^2 - k)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {x^2 (x (4 y (a x y+b)-c x)-2 k)}{4 a}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := x^2*(2*a*x*y+b)*diff(w(x,y),x)- ( 4*a*x^2*y^2 + 3*b*x*y-c*x^2 - k )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (-a \,x^{4} y^{2}-b \,x^{3} y +\frac {1}{4} x^{4} c +\frac {1}{2} k \,x^{2}\right )\]

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6.2.5.35 [511] problem number 35

problem number 511

Added January 2, 2019.

Problem 2.2.5.35 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a x^m w_x + b y^n w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*x^m*D[w[x, y], x] + b*y^n*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {b x^{1-m}}{a (m-1)}-\frac {y^{1-n}}{n-1}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=a*x^m*diff(w(x,y),x)+ b*y^n*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-b \,x^{-m +1} \left (n -1\right )+a \,y^{1-n} \left (m -1\right )}{\left (m -1\right ) a}\right )\]

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6.2.5.36 [512] problem number 36

problem number 512

Added January 2, 2019.

Problem 2.2.5.36 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a x^n w_x + (b y+ c x^m) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*x^n*D[w[x, y], x] + (b*y + c*x^m)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y e^{\frac {b x^{1-n}}{a (n-1)}}-\frac {c (a-a n)^{\frac {-m+n-1}{n-1}} b^{\frac {m-n+1}{n-1}} \Gamma \left (\frac {-m+n-1}{n-1},\frac {b x^{1-n}}{a-a n}\right )}{a (n-1)}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=a*x^n*diff(w(x,y),x)+ (b*y+c*x^m)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {a \,{\mathrm e}^{\frac {b \,x^{-n +1}}{2 a \left (n -1\right )}} \left (-\frac {b \,x^{-n +1}}{a \left (n -1\right )}\right )^{\frac {m -2 n +2}{2 n -2}} c \,x^{m} \left (n -1\right ) \left (m -2 n +2\right )^{2} \operatorname {WhittakerM}\left (\frac {2 n -2-m}{2 n -2}, \frac {-m +3 n -3}{2 n -2}, -\frac {b \,x^{-n +1}}{a \left (n -1\right )}\right )-x^{m} \left (-\frac {b \,x^{-n +1}}{a \left (n -1\right )}\right )^{\frac {m -2 n +2}{2 n -2}} \left (n -1\right )^{2} \left (b \,x^{-n +1}+a \left (m -2 n +2\right )\right ) c \,{\mathrm e}^{\frac {b \,x^{-n +1}}{2 a \left (n -1\right )}} \operatorname {WhittakerM}\left (-\frac {m}{2 n -2}, \frac {-m +3 n -3}{2 n -2}, -\frac {b \,x^{-n +1}}{a \left (n -1\right )}\right )+{\mathrm e}^{\frac {b \,x^{-n +1}}{a \left (n -1\right )}} y a b \left (m -3 n +3\right ) \left (m -n +1\right ) \left (m -2 n +2\right )}{a b \left (m -3 n +3\right ) \left (m -n +1\right ) \left (m -2 n +2\right )}\right )\]

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6.2.5.37 [513] problem number 37

problem number 513

Added January 2, 2019.

Problem 2.2.5.37 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a x^k w_x + (y^n+ b x^m y) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*x^k*D[w[x, y], x] + (y^n + b*x^m*y)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}, Assumptions -> {n != 1}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left ((k-m-1)^{\frac {m}{k-m-1}} a^{\frac {m}{k-m-1}} b^{\frac {1-k}{k-m-1}} (n-1)^{\frac {m}{-k+m+1}} \Gamma \left (\frac {k-1}{k-m-1},\frac {b (n-1) x^{-k+m+1}}{a (k-m-1)}\right )+y^{1-n} e^{-\frac {b (n-1) x^{-k+m+1}}{a (k-m-1)}}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=a*x^k*diff(w(x,y),x)+ (y^n+b*x^m*y)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) assuming n<>1),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\left (-4 x^{-m} x^{k} a \left (\frac {b \,x^{-k +m +1} \left (n -1\right )}{a \left (k -m -1\right )}\right )^{\frac {-2 k +m +2}{2 k -2 m -2}} \left (k -m -1\right ) {\mathrm e}^{-\frac {b \,x^{-k} x^{m} x \left (n -1\right )}{2 a \left (k -m -1\right )}} \left (k -1-\frac {m}{2}\right )^{2} \operatorname {WhittakerM}\left (\frac {2 k -2-m}{2 k -2 m -2}, \frac {3 k -3-2 m}{2 k -2 m -2}, \frac {b \,x^{-k} x^{m} x \left (n -1\right )}{a \left (k -m -1\right )}\right )-2 x^{-m} \left (a \left (k -1-\frac {m}{2}\right ) x^{k}+\frac {x \,x^{m} b \left (n -1\right )}{2}\right ) \left (\frac {b \,x^{-k +m +1} \left (n -1\right )}{a \left (k -m -1\right )}\right )^{\frac {-2 k +m +2}{2 k -2 m -2}} \left (k -m -1\right )^{2} {\mathrm e}^{-\frac {b \,x^{-k} x^{m} x \left (n -1\right )}{2 a \left (k -m -1\right )}} \operatorname {WhittakerM}\left (\frac {m}{2 k -2 m -2}, \frac {3 k -3-2 m}{2 k -2 m -2}, \frac {b \,x^{-k} x^{m} x \left (n -1\right )}{a \left (k -m -1\right )}\right )+6 \left (k -1\right ) x^{k} a \,y^{-n +1} {\mathrm e}^{-\frac {b \,x^{-k +m +1} \left (n -1\right )}{a \left (k -m -1\right )}} \left (k -1-\frac {2 m}{3}\right ) \left (k -1-\frac {m}{2}\right ) b \right ) x^{-k}}{b \left (k -1\right ) \left (3 k -3-2 m \right ) \left (2 k -2-m \right ) a}\right )\]

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6.2.5.38 [514] problem number 38

problem number 514

Added January 2, 2019.

Problem 2.2.5.38 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x(a x^k+b) w_x + \left ( \alpha x^n y^2+(\beta -a n x^k)y+\gamma x^{-n} \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x*(a*x^k + b)*D[w[x, y], x] + (alpha*x^n*y^2 + (beta - a*n*x^k)*y + gamma/x^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (-\frac {\left (\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {(b n+\beta )^2}{\alpha \gamma }-4}+2 \alpha y x^n+b n+\beta \right ) \exp \left (-\frac {\sqrt {\alpha } \sqrt {\gamma } \left (\log \left (a x^k+b\right )+\log (b)-k \log (x)+\log (k)\right ) \sqrt {\frac {(b n+\beta )^2}{\alpha \gamma }-4}}{b k}\right )}{-\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {(b n+\beta )^2}{\alpha \gamma }-4}+2 \alpha y x^n+b n+\beta }\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=x*(a*x^k+b)*diff(w(x,y),x)+ (alpha*x^n*y^2+(beta-a*n*x^k)*y+g*x^(-n))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (-\frac {\left (2 b k \left (b n +\beta \right ) \operatorname {arctanh}\left (\frac {\left (b n +\beta \right ) \left (2 y \alpha \,x^{n}+b n +\beta \right )}{\sqrt {\left (b n +\beta \right )^{2} \left (b^{2} n^{2}+2 b \beta n -4 g \alpha +\beta ^{2}\right )}}\right )+\sqrt {\left (b n +\beta \right )^{2} \left (b^{2} n^{2}+2 b \beta n -4 g \alpha +\beta ^{2}\right )}\, \left (k \ln \left (x \right )-\ln \left (a \,x^{k}+b \right )\right )\right ) \left (b n +\beta \right )}{\sqrt {\left (b n +\beta \right )^{2} \left (b^{2} n^{2}+2 b \beta n -4 g \alpha +\beta ^{2}\right )}\, b k}\right )\]

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6.2.5.39 [515] problem number 39

problem number 515

Added January 2, 2019.

Problem 2.2.5.39 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (y+ A x^n + a) w_x - \left ( n A x^{n-1} y + k x^m + b\right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  (y + A*x^n + a)*D[w[x, y], x] - (n*A*x^(n - 1)*y + k*x^m + b)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y \left (2 a+2 A x^n+y\right )+2 b x+\frac {2 k x^{m+1}}{m+1}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=(y+ A*x^n + a)*diff(w(x,y),x)- ( n*A*x^(n-1)*y + k*x^m + b)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-2 x^{m +1} k -2 \left (A \,x^{n} y +a y +b x +\frac {y^{2}}{2}\right ) \left (m +1\right )}{2 m +2}\right )\]

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6.2.5.40 [516] problem number 40

problem number 516

Added January 2, 2019.

Problem 2.2.5.40 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (y+ a x^{n+1}+b x^n) w_x + \left (a n x^n + c x^{n-1} \right ) y w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (y + a*x^(n + 1) + b*x^n)*D[w[x, y], x] + (a*n*x^n + c*x^(n - 1))*y*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde :=(y+ a*x^(n+1)+b*x^n)*diff(w(x,y),x)+ ( a*n*x^n + c*x^(n-1))*y*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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6.2.5.41 [517] problem number 41

problem number 517

Added January 2, 2019.

Problem 2.2.5.41 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x(2 a x^n y+b) w_x - \left (a(3 n+m)x^n y^2+b(2 n+m)y-A x^m -C x^{-n} \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x*(2*a*x^n*y + b)*D[w[x, y], x] - (a*(3*n + m)*x^n*y^2 + b*(2*n + m)*y - A*x^m - C0/x^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (-\frac {-A x^{m+2 n} \left (2 y (m+n) \left (a y x^n+b\right )-A x^m\right )+2 A \text {C0} x^{m+n}+\text {C0}^2}{2 a A (m+n)}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=x*(2*a*x^n*y+b)*diff(w(x,y),x)- ( a*(3*n+m)*x^n*y^2+b*(2*n+m)*y-A*x^m -C*x^(-n))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {x^{m +n} \left (x^{m +n} A -2 a \,y^{2} \left (m +n \right ) x^{2 n}-2 b y \left (m +n \right ) x^{n}+2 C \right )}{2 m +2 n}\right )\]

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6.2.5.42 [518] problem number 42

problem number 518

Added January 2, 2019.

Problem 2.2.5.42 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a x^n+b x^2+ x y) w_x + \left (c x^n + b x y+ y^2 \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*x^n + b*x^2 + x*y)*D[w[x, y], x] + (c*x^n + b*x*y + y^2)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=(a*x^n+b*x^2+ x*y)*diff(w(x,y),x)+ ( c*x^n + b*x*y+ y^2)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\left (\ln \left (\frac {\left (n^{2}-3 n +3\right ) \left (a^{2} \left (n -1\right ) x^{n}+x \left (\left (b \left (n -1\right ) x +y \left (n -2\right )\right ) a +c x \right )\right )}{\left (2 n -3\right ) \left (a \,x^{n}+b \,x^{2}+x y \right ) a}\right )+\left (-n +1\right ) \ln \left (\frac {\left (n^{2}-3 n +3\right ) \left (x^{n} a^{2}+\left (b a +c \right ) x^{2}\right )}{\left (n -3\right ) a \left (a \,x^{n}+x \left (b x +y \right )\right )}\right )+\left (n -2\right ) \ln \left (-\frac {\left (n^{2}-3 n +3\right ) \left (a y -c x \right ) x}{\left (a \,x^{n}+b \,x^{2}+x y \right ) n a}\right )-2 \left (\ln \left (x \right )-\frac {\ln \left (x^{n} a^{2}+\left (b a +c \right ) x^{2}\right )}{2}\right ) \left (n -1\right )\right ) \left (n^{2}-3 n +3\right )}{3 \left (n -1\right ) \left (n -2\right )}\right )\]

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6.2.5.43 [519] problem number 43

problem number 519

Added January 2, 2019.

Problem 2.2.5.43 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a y^n+b x^2+c x y) w_x + \left (k y^n+ b x y+c y^2\right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*y^n + b*x^2 + c*x*y)*D[w[x, y], x] + (k*y^n + b*x*y + c*y^2)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=(a*y^n+b*x^2+c*x*y)*diff(w(x,y),x)+ ( k*y^n+ b*x*y+c*y^2)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\left (\ln \left (\frac {\left (n^{2}-3 n +3\right ) \left (k^{2} \left (n -1\right ) y^{n}+\left (\left (c \left (n -1\right ) y +b x \left (n -2\right )\right ) k +a b y \right ) y \right )}{\left (2 n -3\right ) \left (k \,y^{n}+b x y +c \,y^{2}\right ) k}\right )+\left (-n +1\right ) \ln \left (\frac {\left (n^{2}-3 n +3\right ) \left (a b \,y^{2}+c k \,y^{2}+y^{n} k^{2}\right )}{\left (n -3\right ) \left (k \,y^{n}+b x y +c \,y^{2}\right ) k}\right )+\left (n -2\right ) \ln \left (\frac {\left (n^{2}-3 n +3\right ) \left (a y -k x \right ) b y}{\left (k \,y^{n}+b x y +c \,y^{2}\right ) k n}\right )+\left (n -1\right ) \left (\ln \left (a b \,y^{2}+c k \,y^{2}+y^{n} k^{2}\right )-2 \ln \left (y \right )\right )\right ) \left (n^{2}-3 n +3\right )}{3 \left (n -1\right ) \left (n -2\right )}\right )\]

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6.2.5.44 [520] problem number 44

problem number 520

Added January 2, 2019.

Problem 2.2.5.44 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a x^n + b x^m + c) w_x + \left (c y^2-b x^{m-1} y+ a x^{n-2}\right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*x^n + b*x^m + c)*D[w[x, y], x] + (c*y^2 - b*x^(m - 1)*y + a*x^(n - 2))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde :=(a*x^n + b*x^m + c)*diff(w(x,y),x)+ ( c*y^2-b*x^(m-1)*y+ a*x^(n-2))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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6.2.5.45 [521] problem number 45

problem number 521

Added January 2, 2019.

Problem 2.2.5.45 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a x^n + b x^m + c) w_x + \left (a x^{n-2} y^2 + b x^{m-1} y + c \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*x^n + b*x^m + c)*D[w[x, y], x] + (a*x^(n - 2)*y^2 + b*x^(m - 1)*y + c)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde :=(a*x^n + b*x^m + c)*diff(w(x,y),x)+ ( a*x^(n-2)*y^2 + b*x^(m-1)*y + c)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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6.2.5.46 [522] problem number 46

problem number 522

Added January 2, 2019.

Problem 2.2.5.46 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a x^n + b x^m + c) w_x + \left ( \alpha x^k y^2 + \beta x^s y - \alpha \lambda ^2 x^k + \beta \lambda x^s \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*x^n + b*x^m + c)*D[w[x, y], x] + (alpha*x^k*y^2 + beta*x^s*y - alpha*lambda^2*x^k + beta*lambda*x^s)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=(a*x^n + b*x^m + c)*diff(w(x,y),x)+ (alpha*x^k*y^2 + beta*x^s*y - alpha*lambda^2*x^k + beta*lambda*x^s)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-\alpha \int \frac {x^{k} {\mathrm e}^{-\int \frac {2 x^{k} \alpha \lambda -x^{s} \beta }{a \,x^{n}+b \,x^{m}+c}d x}}{a \,x^{n}+b \,x^{m}+c}d x \lambda \,{\mathrm e}^{\int \frac {-2 x^{k} \alpha \lambda +x^{s} \beta }{a \,x^{n}+b \,x^{m}+c}d x +\int \frac {2 x^{k} \alpha \lambda -x^{s} \beta }{a \,x^{n}+b \,x^{m}+c}d x}-y \alpha \int \frac {x^{k} {\mathrm e}^{-\int \frac {2 x^{k} \alpha \lambda -x^{s} \beta }{a \,x^{n}+b \,x^{m}+c}d x}}{a \,x^{n}+b \,x^{m}+c}d x -{\mathrm e}^{\int \frac {-2 x^{k} \alpha \lambda +x^{s} \beta }{a \,x^{n}+b \,x^{m}+c}d x}}{\lambda +y}\right )\]

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6.2.5.47 [523] problem number 47

problem number 523

Added January 2, 2019.

Problem 2.2.5.47 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x(a x^n + b x^m + c) w_x - \left ( s x^k y^2 -(a x^n + b x^m+c) y - s \lambda x^{k+2} \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  x*(a*x^n + b*x^m + c)*D[w[x, y], x] - (s*x^k*y^2 - (a*x^n + b*x^m + c)*y - s*lambda*x^(k + 2))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=x*(a*x^n + b*x^m + c)*diff(w(x,y),x)- (s*x^k*y^2 -(a*x^n + b*x^m+c)*y - s*lambda*x^(k+2))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-\int \frac {x^{k}}{a \,x^{n}+b \,x^{m}+c}d x s \sqrt {\lambda }+\operatorname {arctanh}\left (\frac {y}{x \sqrt {\lambda }}\right )}{s \sqrt {\lambda }}\right )\]

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6.2.5.48 [524] problem number 48

problem number 524

Added January 2, 2019.

Problem 2.2.5.48 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a x^n + b x^m + c) w_x + \left ( (a x^n+b x^m + c)y^2-a n(n-1)x^{n-2}-b m(m-1) x^{m-2}\right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  (a*x^n + b*x^m + c)*D[w[x, y], x] + ((a*x^n + b*x^m + c)*y^2 - a*n*(n - 1)*x^(n - 2) - b*m*(m - 1)*x^(m - 2))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (-\frac {\left (a x^n+b x^m+c\right ) \left (a x^n (n+x y)+b x^m (m+x y)+c x y\right )}{\left (a x^n+b x^m+c\right ) \left (a x^n (n+x y)+b x^m (m+x y)+c x y\right ) \int _1^x\frac {1}{\left (b K[1]^m+a K[1]^n+c\right )^2}dK[1]+x}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=(a*x^n + b*x^m + c)*diff(w(x,y),x)+ ((a*x^n+b*x^m + c)*y^2-a*n*(n-1)*x^(n-2)-b*m*(m-1)*x^(m-2))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\left (b \left (m -n \right ) x^{m}-c n \right ) \left (a \,x^{n}+b \,x^{m}+c \right ) \left (b^{2} \left (x y +m \right ) x^{2 m}+x^{2 n} x \,a^{2} y +b \left (a \left (2 x y +m +n \right ) x^{n}+c \left (2 x y +m \right )\right ) x^{m}+n \,x^{2 n} a^{2}+a c \left (2 x y +n \right ) x^{n}+y \,c^{2} x \right ) \int \frac {b \left (m -1+n \right ) \left (m -n \right ) x^{m}-c n \left (n -1\right )}{\left (b \left (m -n \right ) x^{m}-c n \right )^{2} \left (a \,x^{n}+b \,x^{m}+c \right )}d x +\left (b^{2} \left (x y +m \right ) x^{2 m}+x^{2 n} x \,a^{2} y -b^{2} \left (m -n \right ) x^{2 m}+2 b \left (x y +n \right ) \left (a \,x^{n}+c \right ) x^{m}+n \,x^{2 n} a^{2}+2 a c \left (x y +n \right ) x^{n}+c^{2} \left (x y +n \right )\right ) x}{\left (b^{2} \left (x y +m \right ) x^{2 m}+a^{2} \left (x y +n \right ) x^{2 n}+b \left (a \left (2 x y +m +n \right ) x^{n}+c \left (2 x y +m \right )\right ) x^{m}+c \left (a \left (2 x y +n \right ) x^{n}+c y x \right )\right ) \left (b \left (m -n \right ) x^{m}-c n \right ) \left (a \,x^{n}+b \,x^{m}+c \right )}\right )\]

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6.2.5.49 [525] problem number 49

problem number 525

Added January 2, 2019.

Problem 2.2.5.49 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a x^n + b y^n + x) w_x + \left ( \alpha x^k y^{n-k} + \beta x^m y^{n-m} + y \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*x^n + b*y^n + x)*D[w[x, y], x] + (alpha*x^k*y^(n - k) + beta*x^m*y^(n - m) + y)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde := (a*x^n + b*y^n + x)*diff(w(x,y),x)+ (alpha*x^k*y^(n-k) +beta*x^m*y^(n-m) + y )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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6.2.5.50 [526] problem number 50

problem number 526

Added January 2, 2019.

Problem 2.2.5.50 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a x^n + b y^n + A x^2 + B x y) w_x + \left ( \alpha x^k y^{n-k} + \beta x^m y^{n-m} + A x y + B y^2\right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*x^n + b*y^n + A*x^2 + B*x*y)*D[w[x, y], x] + (alpha*x^k*y^(n - k) + beta*x^m*y^(n - m) + A*x*y + B*y^2)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde := (a*x^n + b*y^n + A*x^2 + B*x*y)*diff(w(x,y),x)+ (alpha*x^k*y^(n-k)+beta*x^m*y^(n-m) + A*x*y + B*y^2 )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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6.2.5.51 [527] problem number 51

problem number 527

Added January 2, 2019.

Problem 2.2.5.51 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a y^m + b x^n + s) w_x - \left ( \alpha x^k + b n x^{n-1} y + \beta \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*y^m + b*x^n + s)*D[w[x, y], x] - (alpha*x^k + b*n*x^(n - 1)*y + beta)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := (a*y^m + b*x^n + s)*diff(w(x,y),x)- (alpha*x^k + b*n*x^(n-1)* y + beta)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-\alpha \left (1+m \right ) x^{k +1}-\left (k +1\right ) \left (a \,y^{1+m}+\left (1+m \right ) \left (x^{n} b y +s y +\beta x \right )\right )}{\left (1+m \right ) \left (k +1\right )}\right )\]

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6.2.5.52 [528] problem number 52

problem number 528

Added January 2, 2019.

Problem 2.2.5.52 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a x^n y^m +x) w_x + \left ( b x^k y^{n+m-k} + y \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*x^n*y^m + x)*D[w[x, y], x] + (b*x^k*y^(n + m - k) + y)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde := (a*x^n*y^m +x)*diff(w(x,y),x)+  (b*x^k*y^(n+m-k) + y)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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6.2.5.53 [529] problem number 53

problem number 529

Added January 2, 2019.

Problem 2.2.5.53 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x(a x^n y^m +\alpha ) w_x - y \left ( b x^n y^m + \beta \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  x*(a*x^n*y^m + alpha)*D[w[x, y], x] - y*(b*x^n*y^m + beta)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  x*(a*x^n*y^m +alpha)*diff(w(x,y),x)-  y*( b*x^n*y^m + beta )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (x^{\beta m \left (a n -b m \right )} \left (y^{m}\right )^{\alpha \left (a n -b m \right )} \left (y^{m} \left (a n -b m \right ) x^{n}-\beta m +\alpha n \right )^{-m \left (a \beta -\alpha b \right )}\right )\]

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6.2.5.54 [530] problem number 54

problem number 530

Added January 2, 2019.

Problem 2.2.5.54 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x(a n x^k y^{n+k} + s) w_x - y \left ( b m x^{m+k} y^k + s \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  x*(a*n*x^k*y^(n + k) + s)*D[w[x, y], x] - y*(b*m*x^(m + k)*y^k + s)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde :=  x*(a*n*x^k*y^(n+k) + s)*diff(w(x,y),x)- y*( b*m*x^(m+k)*y^k + s )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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6.2.5.55 [531] problem number 55

problem number 531

Added January 2, 2019.

Problem 2.2.5.55 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a x^n y^m + A x^2 + B x y) w_x + \left ( b x^k y^{n+m-k} + A x y+ B y^2 \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*x^n*y^m + A*x^2 + B*x*y)*D[w[x, y], x] + (b*x^k*y^(n + m - k) + A*x*y + B*y^2)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

$Aborted

Maple ✗

restart; 
pde := (a*x^n*y^m + A*x^2 + B*x*y)*diff(w(x,y),x)+ (b*x^k*y^(n+m-k) + A*x*y+ B*y^2)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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6.2.5.56 [532] problem number 56

problem number 532

Added January 2, 2019.

Problem 2.2.5.56 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a x^n y^m + b x y^k) w_x + \left ( \alpha y^s + \beta \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*x^n*y^m + b*x*y^k)*D[w[x, y], x] + (alpha*y^s + beta)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

$Aborted

Maple ✓

restart; 
pde := (a*x^n*y^m + b*x*y^k)*diff(w(x,y),x)+ (alpha*y^s + beta)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (a \left (n -1\right ) \int \frac {y^{m} {\mathrm e}^{b \left (n -1\right ) \int \frac {y^{k}}{\alpha \,y^{s}+\beta }d y}}{\alpha \,y^{s}+\beta }d y +x^{-n +1} {\mathrm e}^{b \left (n -1\right ) \int \frac {y^{k}}{\alpha \,y^{s}+\beta }d y}\right )\]

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