6.2.3 2.3

6.2.3.1 [461] problem number 1

problem number 461

Added January 2, 2019.

Problem 2.2.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (-\frac {e^{\frac {b x^3}{3}} \left (\left (b x^2+y\right ) \text {HeunT}\left [a^2,-((a-2) b),0,0,b,x\right ]+\text {HeunTPrime}\left [a^2,-((a-2) b),0,0,b,x\right ]\right )}{y \text {HeunT}\left [a^2,-a b,0,0,-b,x\right ]+\text {HeunTPrime}\left [a^2,-a b,0,0,-b,x\right ]}\right )\right \}\right \}\]

But it can’t solve it when assuming \(b>0\) which is strange.

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+ (y^2+b*x^2*y-a^2-a*b*x)*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 {3 \left (\left (x^{2} \operatorname {csgn}\left (b \right ) b -b \,x^{2}-2 y \right ) \operatorname {HeunT}\left (-\frac {a^{2} 3^{{2}/{3}}}{\left (b^{2}\right )^{{1}/{3}}}, -3 \left (a -1\right ) \operatorname {csgn}\left (b \right ), 0, \frac {3^{{2}/{3}} \left (b^{2}\right )^{{1}/{6}} x}{3}\right )-\frac {2 \,3^{{2}/{3}} \left (b^{2}\right )^{{1}/{6}} \operatorname {HeunTPrime}\left (-\frac {a^{2} 3^{{2}/{3}}}{\left (b^{2}\right )^{{1}/{3}}}, -3 \left (a -1\right ) \operatorname {csgn}\left (b \right ), 0, \frac {3^{{2}/{3}} \left (b^{2}\right )^{{1}/{6}} x}{3}\right )}{3}\right ) \operatorname {HeunT}\left (-\frac {a^{2} 3^{{2}/{3}}}{\left (b^{2}\right )^{{1}/{3}}}, -3 \left (a -1\right ) \operatorname {csgn}\left (b \right ), 0, \frac {3^{{2}/{3}} \left (b^{2}\right )^{{1}/{6}} x}{3}\right )}{3 \left (\left (x^{2} \operatorname {csgn}\left (b \right ) b -b \,x^{2}-2 y \right ) \operatorname {HeunT}\left (-\frac {a^{2} 3^{{2}/{3}}}{\left (b^{2}\right )^{{1}/{3}}}, -3 \left (a -1\right ) \operatorname {csgn}\left (b \right ), 0, \frac {3^{{2}/{3}} \left (b^{2}\right )^{{1}/{6}} x}{3}\right )-\frac {2 \,3^{{2}/{3}} \left (b^{2}\right )^{{1}/{6}} \operatorname {HeunTPrime}\left (-\frac {a^{2} 3^{{2}/{3}}}{\left (b^{2}\right )^{{1}/{3}}}, -3 \left (a -1\right ) \operatorname {csgn}\left (b \right ), 0, \frac {3^{{2}/{3}} \left (b^{2}\right )^{{1}/{6}} x}{3}\right )}{3}\right ) \operatorname {HeunT}\left (-\frac {a^{2} 3^{{2}/{3}}}{\left (b^{2}\right )^{{1}/{3}}}, -3 \left (a -1\right ) \operatorname {csgn}\left (b \right ), 0, \frac {3^{{2}/{3}} \left (b^{2}\right )^{{1}/{6}} x}{3}\right ) \int \frac {{\mathrm e}^{\frac {x^{3} \operatorname {csgn}\left (b \right ) b}{3}}}{\operatorname {HeunT}\left (-\frac {a^{2} 3^{{2}/{3}}}{\left (b^{2}\right )^{{1}/{3}}}, -3 \left (a -1\right ) \operatorname {csgn}\left (b \right ), 0, \frac {3^{{2}/{3}} \left (b^{2}\right )^{{1}/{6}} x}{3}\right )^{2}}d x -6 \,{\mathrm e}^{\frac {x^{3} \operatorname {csgn}\left (b \right ) b}{3}}}\right )\]

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6.2.3.2 [462] problem number 2

problem number 462

Added January 2, 2019.

Problem 2.2.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {\sqrt [3]{3} b \Gamma \left (\frac {4}{3},\frac {a x^3}{3}\right )}{a^{4/3}}+\frac {c \Gamma \left (\frac {1}{3},\frac {a x^3}{3}\right )}{3^{2/3} \sqrt [3]{a}}+y e^{-\frac {a x^3}{3}}\right )\right \}\right \}\]

Maple ✓

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

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

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6.2.3.3 [463] problem number 3

problem number 463

Added January 2, 2019.

Problem 2.2.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x +(a x^2 y+b y^3) w_y = 0 \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (-\frac {\frac {3 \Gamma \left (\frac {2}{3}\right ) {\mathrm e}^{\frac {2 a \,x^{3}}{3}} \sqrt {-a \,x^{3} \left (-a \,x^{3}\right )^{{1}/{3}}}}{2}-\frac {3 \left (-a \,x^{3}\right )^{{2}/{3}} \Gamma \left (\frac {2}{3}\right ) {\mathrm e}^{\frac {2 a \,x^{3}}{3}}}{2}+2^{{2}/{3}} b x \,y^{2} \left (-a \,x^{3}\right )^{{1}/{3}} \left (3^{{1}/{3}} \Gamma \left (\frac {1}{3}, -\frac {2 a \,x^{3}}{3}\right ) \Gamma \left (\frac {2}{3}\right )-\frac {2 \,3^{{5}/{6}} \pi }{3}\right )}{3 \left (-a \,x^{3}\right )^{{2}/{3}} y^{2} \Gamma \left (\frac {2}{3}\right )}\right )\]

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6.2.3.4 [464] problem number 4

problem number 464

Added January 2, 2019.

Problem 2.2.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (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 :=  diff(w(x,y),x)+ (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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\left (-b^{2}+4 a \right ) \ln \left (x^{2} \left (a \,x^{2} y^{2}+b x y +1\right )\right )+2 \sqrt {b^{2}-4 a}\, b \,\operatorname {arctanh}\left (\frac {2 a x y +b}{\sqrt {b^{2}-4 a}}\right )-8 \ln \left (x y \right ) \left (-\frac {b^{2}}{4}+a \right )}{-2 b^{2}+8 a}\right )\]

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6.2.3.5 [465] problem number 5

problem number 465

Added January 2, 2019.

Problem 2.2.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x +A(a x+b y+c)^3 y^2 w_y = 0 \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \int _1^y\left (-\frac {b c_1}{a^3 A b x^3+3 a^2 A b (c+b K[6787]) x^2+A b (c+b K[6787])^3+a \left (3 A b x (c+b K[6787])^2+1\right )}-\int _1^x\frac {3 a A b^2 c_1 (c+a K[1]+b K[6787])^2}{\left (a^3 A b K[1]^3+3 a^2 A b (c+b K[6787]) K[1]^2+A b (c+b K[6787])^3+a \left (3 A b K[1] (c+b K[6787])^2+1\right )\right )^2}dK[1]\right )dK[6787]+\int _1^x\frac {A b c_1 (c+b y+a K[1])^3}{A b (c+b y)^3+3 a^2 A b K[1]^2 (c+b y)+a^3 A b K[1]^3+a \left (3 A b K[1] (c+b y)^2+1\right )}dK[1]+c_2\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (A \,b^{4} \textit {\_Z}^{3}+3 A \,b^{3} c \,\textit {\_Z}^{2}+3 A \,b^{2} c^{2} \textit {\_Z} +A b \,c^{3}+a \right )}{\sum }\frac {\ln \left (\frac {\left (y -\textit {\_R} \right ) b +a x}{b}\right )}{\left (\textit {\_R} b +c \right )^{2}}}{3 A b}+x \right )\]

Answer contains RootOf

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6.2.3.6 [466] problem number 6

problem number 466

Added January 2, 2019.

Problem 2.2.3.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x +(a x^4 y^3+(b x^2-1)y+c x) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + (a*x^4*y^3 + (b*x^2 - 1)*y + c*x)*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^4*y^3+(b*x^2-1)*y+c*x)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

Answer contains RootOf

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6.2.3.7 [467] problem number 7

problem number 467

Added January 2, 2019.

Problem 2.2.3.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.2.3.8 [468] problem number 8

problem number 468

Added January 2, 2019.

Problem 2.2.3.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.2.3.9 [469] problem number 9

problem number 469

Added January 2, 2019.

Problem 2.2.3.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

$Aborted

Maple ✓

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

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

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