6.2.30 9.1

6.2.30.1 [799] problem number 1

problem number 799

Added Feb. 7, 2019.

Problem 2.9.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) w_x + g(y) w_y = 0 \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( f \left ( x \right ) \right ) ^{-1}\,{\rm d}x+\int \! \left ( g \left ( y \right ) \right ) ^{-1}\,{\rm d}y \right ) \]

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6.2.30.2 [800] problem number 2

problem number 800

Added Feb. 7, 2019.

Problem 2.9.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (f(x)+g(y)) w_x + f'(x) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (f[x] + g[y])*D[w[x, y], x] + Derivative[1][f][x]*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

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

\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int \!{{\rm e}^{-y}}g \left ( y \right ) \,{\rm d}y+{{\rm e}^{-y}}f \left ( x \right ) \right ) \]

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6.2.30.3 [801] problem number 3

problem number 801

Added Feb. 7, 2019.

Problem 2.9.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

Failed

Maple ✓

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

\[w \left ( x,y \right ) ={\it \_F1} \left ( \left ( n-1 \right ) \int \!{\frac {f \left ( y \right ) }{h \left ( y \right ) }{{\rm e}^{ \left ( n-1 \right ) \int \!{\frac {g \left ( y \right ) }{h \left ( y \right ) }}\,{\rm d}y}}}\,{\rm d}y+{x}^{-n+1}{{\rm e}^{ \left ( n-1 \right ) \int \!{\frac {g \left ( y \right ) }{h \left ( y \right ) }}\,{\rm d}y}} \right ) \]

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6.2.30.4 [802] problem number 4

problem number 802

Added Feb. 7, 2019.

Problem 2.9.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

Failed

Maple ✗

restart; 
pde := (f(y) + a*m*x^n*y^(m-1))*diff(w(x,y),x)-(g(x)+a*n*x^(n-1)*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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6.2.30.5 [803] problem number 5

problem number 803

Added Feb. 7, 2019.

Problem 2.9.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (e^{\alpha x} f(y) + c \beta ) w_x - (e^{\beta y} g(x) + c \alpha ) w_y = 0 \]

Mathematica ✗

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

Failed

Maple ✗

restart; 
pde := (exp(alpha*x)* f(y) + c*beta)*diff(w(x,y),x)-(exp(beta*y)*g(x) + c*alpha)*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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