6.2.2 2.2

6.2.2.1 [430] problem number 1

problem number 430

Added January 2, 2019.

Problem 2.2.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.2 [431] problem number 2

problem number 431

Added January 2, 2019.

Problem 2.2.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.3 [432] problem number 3

problem number 432

Added January 2, 2019.

Problem 2.2.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.4 [433] problem number 4

problem number 433

Added January 2, 2019.

Problem 2.2.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\frac {-\frac {{\mathrm e}^{\frac {k^{2}}{2 a}} \left (a^{2} s +\left (-k c +b \right ) a +b \,k^{2}\right ) \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {2}\, \left (a x +k \right )}{2 \sqrt {a}}\right )}{2}+\sqrt {a}\, \left (a^{2} y +\left (b x +c \right ) a -k b \right ) {\mathrm e}^{-\frac {x \left (a x +2 k \right )}{2}}}{a^{{5}/{2}}}\right )\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.5 [434] problem number 5

problem number 434

Added January 2, 2019.

Problem 2.2.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {\operatorname {ParabolicCylinderD}\left (-2,i \sqrt {2} \sqrt {a} x\right ) (a x-y)+i \sqrt {2} \sqrt {a} \operatorname {ParabolicCylinderD}\left (-1,i \sqrt {2} \sqrt {a} x\right )}{\operatorname {ParabolicCylinderD}\left (1,\sqrt {2} \sqrt {a} x\right ) (a x+y)-\sqrt {2} \sqrt {a} \operatorname {ParabolicCylinderD}\left (2,\sqrt {2} \sqrt {a} x\right )}\right )\right \}\right \}\]

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.6 [435] problem number 6

problem number 435

Added January 2, 2019.

Problem 2.2.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {a} x\right ) (y-a x)+2 \sqrt {a} e^{a x^2}}{2 \sqrt {a} (a x-y)}\right )\right \}\right \}\]

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.7 [436] problem number 7

problem number 436

Added January 2, 2019.

Problem 2.2.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a} x}{\sqrt {2}}\right )-\frac {y e^{\frac {a x^2}{2}}}{\sqrt {2} \sqrt {a} (x y+1)}\right )\right \}\right \}\]

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.8 [437] problem number 8

problem number 437

Added January 2, 2019.

Problem 2.2.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {(y-b) \int _1^xe^{\frac {1}{2} K[1] (4 b+a K[1])}dK[1]+e^{\frac {1}{2} x (a x+4 b)}}{b-y}\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (-\frac {\left ({\mathrm e}^{\frac {x \left (a x +4 b \right )}{2}} \sqrt {2}\, \sqrt {-a}+{\mathrm e}^{-\frac {2 b^{2}}{a}} \operatorname {erf}\left (\frac {\left (a x +2 b \right ) \sqrt {2}}{2 \sqrt {-a}}\right ) \sqrt {\pi }\, \left (b -y \right )\right ) \sqrt {2}}{\sqrt {-a}\, \left (2 b -2 y \right )}\right )\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.9 [438] problem number 9

problem number 438

Added January 2, 2019.

Problem 2.2.2.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.10 [439] problem number 10

problem number 439

Added January 2, 2019.

Problem 2.2.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.11 [440] problem number 11

problem number 440

Added January 2, 2019.

Problem 2.2.2.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.12 [441] problem number 12

problem number 441

Added January 2, 2019.

Problem 2.2.2.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {1}{2} \left (2 \arctan \left (\frac {2 \alpha (a y+b x)+\beta }{a \alpha \sqrt {\frac {4 a \alpha \gamma -a \beta ^2+4 \alpha b c}{a^3 \alpha ^2}}}\right )-a \alpha \log (a x+c) \sqrt {\frac {4 a \alpha \gamma -a \beta ^2+4 \alpha b c}{a^3 \alpha ^2}}\right )\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\frac {-2 \arctan \left (\frac {a^{2} \left (2 y a \alpha +2 b x \alpha +\beta \right )}{\sqrt {\left (4 g \alpha -\beta ^{2}\right ) a^{4}+4 a^{3} \alpha b c}}\right ) a^{2}+\ln \left (a x +c \right ) \sqrt {\left (4 g \alpha -\beta ^{2}\right ) a^{4}+4 a^{3} \alpha b c}}{2 a^{2}}\right )\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.13 [442] problem number 13

problem number 442

Added January 2, 2019.

Problem 2.2.2.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

restart; 
pde := a*x^2*diff(w(x,y),x)+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'));
 

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.14 [443] problem number 14

problem number 443

Added January 2, 2019.

Problem 2.2.2.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

restart; 
pde := (a*x^2+b)*diff(w(x,y),x)-(y^2-2*x*y+(1-a)*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'));
 

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.15 [444] problem number 15

problem number 444

Added January 2, 2019.

Problem 2.2.2.15 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a_1 x^2+b_1 x + x_1) w_x + (a_2 y^2+b_2 y+c_2) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  (a1*x^2 + b1*x + c1)*D[w[x, y], x] + (a2*y^2 + b2*y + c2)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\int _1^y\frac {1}{\text {c2}+K[1] (\text {b2}+\text {a2} K[1])}dK[1]-\int _1^x\frac {1}{\text {c1}+K[2] (\text {b1}+\text {a1} K[2])}dK[2]\right )\right \}\right \}\]

Maple ✓

restart; 
pde := (a1*x^2+b1*x+c1)*diff(w(x,y),x)+ (a2*y^2+b2*y+c2)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {2 \arctan \left (\frac {2 \operatorname {a2} y +\operatorname {b2}}{\sqrt {4 \operatorname {c2} \operatorname {a2} -\operatorname {b2}^{2}}}\right ) \sqrt {4 \operatorname {c1} \operatorname {a1} -\operatorname {b1}^{2}}-2 \arctan \left (\frac {2 \operatorname {a1} x +\operatorname {b1}}{\sqrt {4 \operatorname {c1} \operatorname {a1} -\operatorname {b1}^{2}}}\right ) \sqrt {4 \operatorname {c2} \operatorname {a2} -\operatorname {b2}^{2}}}{\sqrt {4 \operatorname {c1} \operatorname {a1} -\operatorname {b1}^{2}}\, \sqrt {4 \operatorname {c2} \operatorname {a2} -\operatorname {b2}^{2}}}\right )\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.16 [445] problem number 16

problem number 445

Added January 2, 2019.

Problem 2.2.2.16 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {1}{2} \left (-\sqrt {-\frac {k^2 (a-b)^2}{(k+1)^2}} \int _1^x\frac {k+1}{(a-K[5]) (K[5]-b)}dK[5]-2 \arctan \left (\frac {a k+b k-2 (k (x+y)+y)}{(k+1) \sqrt {-\frac {k^2 (a-b)^2}{(k+1)^2}}}\right )\right )\right )\right \}\right \}\]

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.17 [446] problem number 17

problem number 446

Added January 2, 2019.

Problem 2.2.2.17 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a_1 y^2+b_1 y + c_1) w_x +( a_2 x^2+b_2 x+c_2) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  (a1*y^2 + b1*y + c1)*D[w[x, y], x] + (a2*x^2 + b2*x + c2)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {1}{6} \left (2 \text {a1} y^3-2 \text {a2} x^3+3 \text {b1} y^2-3 \text {b2} x^2+6 \text {c1} y-6 \text {c2} x\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde := (a1*y^2+b1*y+c1)*diff(w(x,y),x)+ (a2*x^2+b2*x+c2)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {1}{3} \operatorname {a1} \,y^{3}-\frac {1}{3} \operatorname {a2} \,x^{3}+\frac {1}{2} \operatorname {b1} \,y^{2}-\frac {1}{2} \operatorname {b2} \,x^{2}+\operatorname {c1} y -\operatorname {c2} x \right )\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.18 [447] problem number 18

problem number 447

Added January 2, 2019.

Problem 2.2.2.18 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.19 [448] problem number 19

problem number 448

Added January 2, 2019.

Problem 2.2.2.19 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {a y^3}{3}+b x y+\frac {x^4}{4}\right )\right \}\right \}\]

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.20 [449] problem number 20

problem number 449

Added January 2, 2019.

Problem 2.2.2.20 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

Failed

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.21 [450] problem number 21

problem number 450

Added January 2, 2019.

Problem 2.2.2.21 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

Failed

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.22 [451] problem number 22

problem number 451

Added January 2, 2019.

Problem 2.2.2.22 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.23 [452] problem number 23

problem number 452

Added January 2, 2019.

Problem 2.2.2.23 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.24 [453] problem number 24

problem number 453

Added January 2, 2019.

Problem 2.2.2.24 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

Failed

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.25 [454] problem number 25

problem number 454

Added January 2, 2019.

Problem 2.2.2.25 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (A x y+B x^2+k x) w_x +(D y^2+E x y+F x^2+k y)w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (A*x*y + B*x^2 + k*x)*D[w[x, y], x] + (D0*y^2 + E0*x*y + F*x^2 + k*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*y+B*x^2+k*x)*diff(w(x,y),x)+(D0*y^2+E0*x*y+F0*x^2+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=()

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.26 [455] problem number 26

problem number 455

Added January 2, 2019.

Problem 2.2.2.26 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (A x y+A k y+B x^2+B k x) w_x +(C y^2+D x y+k(D-B)y) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (A*x*y + A*k*y + B*x^2 + B*k*x)*D[w[x, y], x] + (C0*y^2 + D0*x*y + k*(D0 - B)*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*y+A*k*y+B*x^2+B*k*x)*diff(w(x,y),x)+(C0*y^2+D0*x*y+k*(D0-B)*y)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.27 [456] problem number 27

problem number 456

Added January 2, 2019.

Problem 2.2.2.27 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (A y^2+B x y+C x^2+k x) w_x +(D y^2+E x y + F x^2+k y) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (A*y^2 + B*x*y + C0*x^2 + k*x)*D[w[x, y], x] + (D0*y^2 + E0*x*y + F0*x^2 + 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*y^2+B*x*y+C0*x^2+k*x)*diff(w(x,y),x)+(D0*y^2+E0*x*y+F0*x^2+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=()

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.28 [457] problem number 28

problem number 457

Added January 2, 2019.

Problem 2.2.2.28 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (A y^2+B x y+C x^2) w_x +(D y^2+E x y + F x^2) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (A*y^2 + B*x*y + C0*x^2)*D[w[x, y], x] + (D0*y^2 + E0*x*y + F0*x^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^2+B*x*y+C0*x^2)*diff(w(x,y),x)+(D0*y^2+E0*x*y+F0*x^2)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (-\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (A \,\textit {\_Z}^{3}+\left (B -\operatorname {D0} \right ) \textit {\_Z}^{2}+\left (\operatorname {C0} -\operatorname {E0} \right ) \textit {\_Z} -\operatorname {F0} \right )}{\sum }\frac {\left (A \,\textit {\_R}^{2}+B \textit {\_R} +\operatorname {C0} \right ) \ln \left (\frac {-\textit {\_R} x +y}{x}\right )}{3 A \,\textit {\_R}^{2}+2 \left (B -\operatorname {D0} \right ) \textit {\_R} +\operatorname {C0} -\operatorname {E0}}\right )-\ln \left (x \right )\right )\]

solution contains RootOf

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.29 [458] problem number 29

problem number 458

Added January 2, 2019.

Problem 2.2.2.29 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (A*y^2 + 2*B*x*y + D0*x^2 + a)*D[w[x, y], x] - (D0*y^2 + 2*D0*x*y - E0*x^2 - b)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde :=  (A*y^2+2*B*x*y+D0*x^2+a)*diff(w(x,y),x)-(D0*y^2+2*D0*x*y-E0*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'));
 

sol=()

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.30 [459] problem number 30

problem number 459

Added January 2, 2019.

Problem 2.2.2.30 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

Failed

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.2.2.31 [460] problem number 31, Hesse’s equation

problem number 460

Added January 2, 2019.

Problem 2.2.2.31 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux. Reference E. Kamke (1965).

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

\[ (x f_1-f_2) w_x +(y f_1-f_3)w_y = 0 \]

Where \(f_n = a_n+b_n x + c_n y\).

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (x*(a1 + b1*x + c1*y) - (a2 + b2*x + c2*y))*D[w[x, y], x] + (y*(a1 + b1*x + c1*y) - (a3 + b3*x + c3*y))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  (x *(a1+b1*x+c1*y)-(a2+b2*x+c2*y))*diff(w(x,y),x)+(y*(a1+b1*x+c1*y)-(a3+b3*x+c3*y))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[\text {Expression too large to display}\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________