6.2.31 9.2

6.2.31.1 [804] problem number 1

problem number 804

Added Feb. 7, 2019.

Problem 2.9.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \int _1^y\left (-\frac {b c_1}{a+b f(c+a x+b K[6620])}-\int _1^x\frac {a b^2 c_1 f'(c+a K[1]+b K[6620])}{(a+b f(c+a K[1]+b K[6620]))^2}dK[1]\right )dK[6620]+\int _1^x\frac {b c_1 f(c+b y+a K[1])}{a+b f(c+b y+a K[1])}dK[1]+c_2\right \}\right \}\]

Maple ✓

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

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

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6.2.31.2 [805] problem number 2

problem number 805

Added Feb. 7, 2019.

Problem 2.9.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + f(\frac {y}{x}) w_y = 0 \]

Mathematica ✗

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

\[w \left (x , y\right ) = f_{1} \left (\int _{}^{\frac {y}{x}}\frac {1}{f \left (\textit {\_a} \right )-\textit {\_a}}d \textit {\_a} -\ln \left (x \right )\right )\]

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6.2.31.3 [806] problem number 3

problem number 806

Added Feb. 7, 2019.

Problem 2.9.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (f[y + a*x^n + b] - a*n*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 :=  diff(w(x,y),x)+ (f(y+a*x^n+b) - a*n*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'));
 

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

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6.2.31.4 [807] problem number 4

problem number 807

Added Feb. 7, 2019.

Problem 2.9.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y f(x^n y^m) w_y = 0 \]

Mathematica ✗

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

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

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6.2.31.5 [808] problem number 5

problem number 808

Added Feb. 7, 2019.

Problem 2.9.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

Failed

Maple ✗

restart; 
pde := y^(m-1)*diff(w(x,y),x)+  x^(n-1)*f(a*x^n+b*y^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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6.2.31.6 [809] problem number 6

problem number 809

Added Feb. 7, 2019.

Problem 2.9.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + e^{-\lambda x} f(e^{\lambda x} y) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + Exp[-(lambda*x)]*f[Exp[lambda*x]*y]*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)+   exp(-lambda*x)*f(exp(lambda*x)*y)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (x -\int _{}^{{\mathrm e}^{\lambda x} y}\frac {1}{\textit {\_a} \lambda +f \left (\textit {\_a} \right )}d \textit {\_a} \right )\]

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6.2.31.7 [810] problem number 7

problem number 810

Added Feb. 7, 2019.

Problem 2.9.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + e^{\lambda y} f(e^{\lambda y} x) w_y = 0 \]

Mathematica ✗

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

\[w \left (x , y\right ) = f_{1} \left (\frac {-\int _{}^{\frac {\lambda y +\ln \left (x \right )}{\lambda }}\frac {1}{1+f \left ({\mathrm e}^{\lambda \textit {\_a}}\right ) \lambda \,{\mathrm e}^{\lambda \textit {\_a}}}d \textit {\_a} \lambda +\ln \left (x \right )}{\lambda }\right )\]

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6.2.31.8 [811] problem number 8

problem number 811

Added Feb. 7, 2019.

Problem 2.9.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + y f(e^{\alpha x} y^m) w_y = 0 \]

Mathematica ✗

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

\[w \left (x , y\right ) = f_{1} \left (\frac {\int _{\textit {\_b}}^{y}\frac {1}{\textit {\_a} \left (f \left ({\mathrm e}^{\alpha x} \textit {\_a}^{m}\right ) m +\alpha \right )}d \textit {\_a} m -x}{m}\right )\]

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6.2.31.9 [812] problem number 9

problem number 812

Added Feb. 7, 2019.

Problem 2.9.2.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + f(x^n e^{\alpha y}) w_y = 0 \]

Mathematica ✗

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

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

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6.2.31.10 [813] problem number 10

problem number 813

Added Feb. 7, 2019.

Problem 2.9.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + e^{\lambda x - \beta y} f(a e^{\lambda x} + b e^{\beta y}) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + Exp[lambda*x - beta*y]*f[a*Exp[lambda*x] + b*Exp[beta*y]]*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)+  exp(lambda*x-beta*y)*f(a*exp(lambda*x)+b*exp(beta*y))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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6.2.31.11 [814] problem number 11

problem number 814

Added Feb. 7, 2019.

Problem 2.9.2.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

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

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6.2.31.12 [815] problem number 12

problem number 815

Added Feb. 7, 2019.

Problem 2.9.2.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \alpha x y w_x + \left ( \alpha f(x^n e^{\alpha y}) - n y \right ) w_y = 0 \]

Mathematica ✗

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

Failed

Maple ✗

restart; 
pde :=  alpha*x*y*diff(w(x,y),x)+  (alpha*f(x^n*exp(alpha*y)) - n*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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6.2.31.13 [816] problem number 13

problem number 816

Added Feb. 7, 2019.

Problem 2.9.2.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ m x(\ln y) w_x + \left ( y f(x^n y^m) - n y \ln y \right ) w_y = 0 \]

Mathematica ✗

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

Failed

Maple ✗

restart; 
pde :=   m*x*ln(y)*diff(w(x,y),x)+  (y*f(x^n*y^m) - n*y*ln[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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6.2.31.14 [817] problem number 14

problem number 817

Added Feb. 7, 2019.

Problem 2.9.2.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

\[w \left (x , y\right ) = f_{1} \left (-\arctan \left (\tan \left (x \right )\right )-\int _{}^{-a \tan \left (x \right )-y}\frac {1}{a +f \left (-\textit {\_a} \right )}d \textit {\_a} \right )\]

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6.2.31.15 [818] problem number 15

problem number 818

Added Feb. 7, 2019.

Problem 2.9.2.15 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ e^{\lambda x} w_x + f(\lambda x+\ln y) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  Exp[lambda*x]*D[w[x, y], x] + f[lambda*x + Log[y]]*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (x -\int _{}^{y \,{\mathrm e}^{\lambda x}}\frac {1}{f \left (\ln \left (\textit {\_a} \right )\right )+\textit {\_a} \lambda }d \textit {\_a} \right )\]

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6.2.31.16 [819] problem number 16

problem number 819

Added Feb. 7, 2019.

Problem 2.9.2.16 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + e^{\lambda y} f(\lambda y+\ln x) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + Exp[lambda*y]*f[lambda*y + Log[x]]*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)+  exp(lambda*y)*f(lambda*y+ln(x))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-\int _{}^{\frac {\lambda y +\ln \left (x \right )}{\lambda }}\frac {1}{1+f \left (\lambda \textit {\_a} \right ) \lambda \,{\mathrm e}^{\lambda \textit {\_a}}}d \textit {\_a} \lambda +\ln \left (x \right )}{\lambda }\right )\]

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