6.2.28 8.5

6.2.28.1 [782] problem number 1

problem number 782

Added Feb. 7, 2019.

Problem 2.8.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( \lambda \sin (\lambda x) y^2 + f(x) \cos (\lambda x) y-f(x) \right ) w_y = 0 \]

Mathematica ✗

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

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {y \cos \left (\lambda x \right )-1}{-y \left (\int -\lambda \,{\mathrm e}^{\int \frac {-2 \lambda \left (\cos ^{2}\left (\lambda x \right )\right )+\sqrt {-\frac {\cos \left (2 \lambda x \right )}{2}+\frac {1}{2}}\, \left (\cos ^{2}\left (\lambda x \right )\right ) f \left (x \right )+2 \lambda }{\cos \left (\lambda x \right ) \sin \left (\lambda x \right )}d x} \sin \left (\lambda x \right )d x \right ) \cos \left (\lambda x \right )+\cos \left (\lambda x \right ) {\mathrm e}^{\int \frac {-2 \lambda \left (\cos ^{2}\left (\lambda x \right )\right )+\sqrt {-\frac {\cos \left (2 \lambda x \right )}{2}+\frac {1}{2}}\, \left (\cos ^{2}\left (\lambda x \right )\right ) f \left (x \right )+2 \lambda }{\cos \left (\lambda x \right ) \sin \left (\lambda x \right )}d x}+\int -\lambda \,{\mathrm e}^{\int \frac {-2 \lambda \left (\cos ^{2}\left (\lambda x \right )\right )+\sqrt {-\frac {\cos \left (2 \lambda x \right )}{2}+\frac {1}{2}}\, \left (\cos ^{2}\left (\lambda x \right )\right ) f \left (x \right )+2 \lambda }{\cos \left (\lambda x \right ) \sin \left (\lambda x \right )}d x} \sin \left (\lambda x \right )d x}\right )\]

____________________________________________________________________________________

6.2.28.2 [783] problem number 2

problem number 783

Added Feb. 7, 2019.

Problem 2.8.5.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( f(x) y^2-a^2 f(x)+a \lambda \sin (\lambda x)+a^2 f(x) \sin ^2(\lambda x) \right ) w_y = 0 \]

Mathematica ✗

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

____________________________________________________________________________________

6.2.28.3 [784] problem number 3

problem number 784

Added Feb. 7, 2019.

Problem 2.8.5.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( f(x) y^2-a^2 f(x)+a \lambda \cos (\lambda x)+a^2 f(x) \cos ^2(\lambda x) \right ) w_y = 0 \]

Mathematica ✗

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

____________________________________________________________________________________

6.2.28.4 [785] problem number 4

problem number 785

Added Feb. 7, 2019.

Problem 2.8.5.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

sol=()

____________________________________________________________________________________

6.2.28.5 [786] problem number 5

problem number 786

Added Feb. 7, 2019.

Problem 2.8.5.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

sol=()

____________________________________________________________________________________