6.5.24 8.2

6.5.24.1 [1355] Problem 1

problem number 1355

Added April 13, 2019.

Problem Chapter 5.8.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c w + f(x)g(x) \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.5.24.2 [1356] Problem 2

problem number 1356

Added April 13, 2019.

Problem Chapter 5.8.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac { \left ( ay-b \left ( x-{\it \_a} \right ) \right ) g \left ( {\it \_a} \right ) +f \left ( {\it \_a} \right ) {\it \_a}\,a}{{a}^{2}}{{\rm e}^{-{\frac {c{\it \_a}}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{{\frac {cx}{a}}}}\]

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6.5.24.3 [1357] Problem 3

problem number 1357

Added April 13, 2019.

Problem Chapter 5.8.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = f(x) w + g(x) h(x) \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.5.24.4 [1358] Problem 4

problem number 1358

Added April 13, 2019.

Problem Chapter 5.8.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = (f(x)+g(y)) w + p(x)+q(y) \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a}{{\rm e}^{-{\frac {1}{a}\int \!f \left ( {\it \_b} \right ) +g \left ( {\frac {ay-b \left ( x-{\it \_b} \right ) }{a}} \right ) \,{\rm d}{\it \_b}}}} \left ( p \left ( {\it \_b} \right ) +q \left ( {\frac {ay-b \left ( x-{\it \_b} \right ) }{a}} \right ) \right ) }{d{\it \_b}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{a} \left ( f \left ( {\it \_a} \right ) +g \left ( {\frac {ay-b \left ( x-{\it \_a} \right ) }{a}} \right ) \right ) }{d{\it \_a}}}}\]

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6.5.24.5 [1359] Problem 5

problem number 1359

Added April 13, 2019.

Problem Chapter 5.8.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {f \left ( {\it \_a} \right ) }{a}{{\it \_a}}^{{\frac {-a-c}{a}}}g \left ( y{x}^{-{\frac {b}{a}}}{{\it \_a}}^{{\frac {b}{a}}} \right ) }{d{\it \_a}}+{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) \right ) {x}^{{\frac {c}{a}}}\]

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6.5.24.6 [1360] Problem 6

problem number 1360

Added April 13, 2019.

Problem Chapter 5.8.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f_1(x) w_x + f_2(y) w_y = a w + g_1(x)+g_2(y) \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  f1[x]*D[w[x, y], x] + f2[y]*D[w[x, y], y] == a*w[x,y]+g1[x]+g2[y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  f1(x)*diff(w(x,y),x)+ f2(y)*diff(w(x,y),y) = a*w(x,y)+g1(x)+g2(y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {{{\rm e}^{-a\int \! \left ( {\it f1} \left ( {\it \_f} \right ) \right ) ^{-1}\,{\rm d}{\it \_f}}} \left ( {\it g1} \left ( {\it \_f} \right ) +{\it g2} \left ( \RootOf \left ( \int \! \left ( {\it f1} \left ( {\it \_f} \right ) \right ) ^{-1}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\! \left ( {\it f2} \left ( {\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}-\int \! \left ( {\it f1} \left ( x \right ) \right ) ^{-1}\,{\rm d}x+\int \! \left ( {\it f2} \left ( y \right ) \right ) ^{-1}\,{\rm d}y \right ) \right ) \right ) }{{\it f1} \left ( {\it \_f} \right ) }}{d{\it \_f}}+{\it \_F1} \left ( -\int \! \left ( {\it f1} \left ( x \right ) \right ) ^{-1}\,{\rm d}x+\int \! \left ( {\it f2} \left ( y \right ) \right ) ^{-1}\,{\rm d}y \right ) \right ) {{\rm e}^{\int \!{\frac {a}{{\it f1} \left ( x \right ) }}\,{\rm d}x}}\]

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