6.2.25 8.2

6.2.25.1 [764] problem number 1

problem number 764

Added Feb. 4, 2019.

Problem 2.8.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( a e^{\lambda x} y^2 + a e^{\lambda x} f(x) y+\lambda f(x) \right ) w_y = 0 \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.2.25.2 [765] problem number 2

problem number 765

Added Feb. 4, 2019.

Problem 2.8.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {y \exp \left (-\int _1^{e^{\lambda x}}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]-\lambda x\right )}{a e^{\lambda x}-y}-\int _1^{e^{\lambda x}}\frac {\exp \left (-\int _1^{K[2]}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]\right )}{K[2]^2}dK[2]\right )\right \}\right \}\]

Maple ✗

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

sol=()

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6.2.25.3 [766] problem number 3

problem number 766

Added Feb. 4, 2019.

Problem 2.8.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (f[x]*y^2 + a*lambda*Exp[lambda*x] - a^2*Exp[2*lambda*x]*f[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(x)*y^2+a*lambda*exp(lambda*x)-a^2*exp(2*lambda*x)*f(x))*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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6.2.25.4 [767] problem number 4

problem number 767

Added Feb. 4, 2019.

Problem 2.8.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f(x) y^2+\lambda y+ a e^{2 \lambda x} f(x)\right ) w_y = 0 \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\arctan \left (\frac {y e^{-\lambda x}}{\sqrt {a}}\right )-\sqrt {a} \int _1^xe^{\lambda K[1]} f(K[1])dK[1]\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\sqrt {a}\, \int f \left (x \right ) {\mathrm e}^{\lambda x}d x -\arctan \left (\frac {{\mathrm e}^{-\lambda x} y}{\sqrt {a}}\right )\right )\]

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6.2.25.5 [768] problem number 5

problem number 768

Added Feb. 4, 2019.

Problem 2.8.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (f[x]*y^2 - (a*Exp[lambda*x] + b)*f[x]*y + 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(x)*y^2-(a*exp(lambda*x)+b)*f(x)*y+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'));
 

sol=()

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6.2.25.6 [769] problem number 6

problem number 769

Added Feb. 4, 2019.

Problem 2.8.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

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

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6.2.25.7 [770] problem number 7

problem number 770

Added Feb. 4, 2019.

Problem 2.8.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left (f(x) y^2+ g(x) y+a \lambda e^{\lambda x} -a e^{\lambda x} g(x) -a^2 e^{2 \lambda x} f(x)\right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (f[x]*y^2 + g[x]*y + a*lambda*Exp[lambda*x] - a*Exp[lambda*x]*g[x] - a^2*Exp[2*lambda*x]*f[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(x)*y^2+ g(x)*y+a*lambda*exp(lambda*x) -a*exp(lambda*x)*g(x)-a^2*exp(2*lambda*x)*f(x))*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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6.2.25.8 [771] problem number 8

problem number 771

Added Feb. 7, 2019.

Problem 2.8.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left (f(x) y^2- a e^{\lambda x} g(x) y + a \lambda e^{\lambda x} +a^2 e^{2 \lambda x} (g(x)-f(x))\right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (f[x]*y^2 - a*Exp[lambda*x]*g[x]*y + a*lambda*Exp[lambda*x] + a^2*Exp[2*lambda*x]*(g[x] - f[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(x)*y^2- a*exp(lambda*x)*g(x)*y + a*lambda*exp(lambda*x) +a^2*exp(2*lambda*x)* (g(x)-f(x)))*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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6.2.25.9 [772] problem number 9

problem number 772

Added Feb. 7, 2019.

Problem 2.8.2.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (f[x]*y^2 + 2*a*lambda*x*Exp[lambda*x^2] - a^2*f[x]*Exp[2*lambda*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(x)*y^2+2*a*lambda*x*exp(lambda*x^2) - a^2*f(x)*exp(2*lambda*x^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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6.2.25.10 [773] problem number 10

problem number 773

Added Feb. 7, 2019.

Problem 2.8.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left (f(x) y^2+2 \lambda x y+ a f(x) e^{2 \lambda x^2} \right ) w_y = 0 \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\arctan \left (\frac {y e^{-\lambda x^2}}{\sqrt {a}}\right )-\sqrt {a} \int _1^xe^{\lambda K[1]^2} f(K[1])dK[1]\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\sqrt {a}\, \int f \left (x \right ) {\mathrm e}^{\lambda \,x^{2}}d x -\arctan \left (\frac {{\mathrm e}^{-\lambda \,x^{2}} y}{\sqrt {a}}\right )\right )\]

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6.2.25.11 [774] problem number 11

problem number 774

Added Feb. 7, 2019.

Problem 2.8.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

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

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