6.2.28 8.5

6.2.28.1 [783] problem number 1

problem number 783

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'));
 

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

____________________________________________________________________________________

6.2.28.2 [784] problem number 2

problem number 784

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 [785] problem number 3

problem number 785

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 [786] problem number 4

problem number 786

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 [787] problem number 5

problem number 787

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=()

____________________________________________________________________________________