Added April 3, 2019.
Problem Chapter 5.4.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b w_y = c w + \tanh ^k(\lambda x) \tanh ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+Tanh[lambda*x]^k*Tanh[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} \tanh ^k(\lambda K[1]) \tanh ^n\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+tanh(lambda*x)^k*tanh(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac { \left ( \tanh \left ( {\it \_a}\,\lambda \right ) \right ) ^{k}}{a} \left ( \tanh \left ( {\frac {\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n}{{\rm e}^{-{\frac {c{\it \_a}}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{{\frac {cx}{a}}}}\]
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Added April 3, 2019.
Problem Chapter 5.4.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b w_y = c \tanh ^k(\lambda x) w + s \tanh ^n(\beta x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*Tanh[lambda*x]^k*w[x,y]+ s*Tanh[beta*x]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\frac {c \tanh ^{k+1}(\lambda x) \, _2F_1\left (1,\frac {k+1}{2};\frac {k+3}{2};\tanh ^2(\lambda x)\right )}{a k \lambda +a \lambda }\right ) \left (\int _1^x\frac {\exp \left (-\frac {c \, _2F_1\left (1,\frac {k+1}{2};\frac {k+3}{2};\tanh ^2(\lambda K[1])\right ) \tanh ^{k+1}(\lambda K[1])}{a \lambda +a k \lambda }\right ) s \tanh ^n(\beta K[1])}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*tanh(lambda*x)^k*w(x,y)+s*tanh(beta*x)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int \!{\frac {s \left ( \tanh \left ( \beta \,x \right ) \right ) ^{n}}{a}{{\rm e}^{-{\frac {c\int \! \left ( \tanh \left ( x\lambda \right ) \right ) ^{k}\,{\rm d}x}{a}}}}}\,{\rm d}x+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{\int \!{\frac { \left ( \tanh \left ( x\lambda \right ) \right ) ^{k}c}{a}}\,{\rm d}x}}\]
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Added April 3, 2019.
Problem Chapter 5.4.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b w_y = \left (c_1 \tanh ^{n_1}(\lambda _1 x)+ c_2 \tanh ^{n_2}(\lambda _2 y) \right ) w + s_1 \tanh ^{k_1}(\beta _1 x)+ s_2 \tanh ^{k_2}(\beta _2 y) \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == (c1*Tanh[lambda1*x]^n1 + c2*Tanh[lambda2*y]^n2)*w[x,y] + s1*Tanh[beta1*x]^k1+ s2*Tanh[beta2*y]^k2; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
$Aborted
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = (c1*tanh(lambda1*x)^n1 + c2*tanh(lambda2*y)^n2)*w(x,y) + s1*tanh(beta1*x)^k1+ s2*tanh(beta2*y)^k2; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a}{{\rm e}^{-{\frac {1}{a}\int \!{\it c1}\, \left ( \tanh \left ( \lambda 1\,{\it \_b} \right ) \right ) ^{{\it n1}}+{\it c2}\, \left ( \tanh \left ( {\frac {\lambda 2\, \left ( ay-b \left ( x-{\it \_b} \right ) \right ) }{a}} \right ) \right ) ^{{\it n2}}\,{\rm d}{\it \_b}}}} \left ( {\it s1}\, \left ( \tanh \left ( \beta 1\,{\it \_b} \right ) \right ) ^{{\it k1}}+{\it s2}\, \left ( \tanh \left ( {\frac {\beta 2\, \left ( ay-b \left ( x-{\it \_b} \right ) \right ) }{a}} \right ) \right ) ^{{\it k2}} \right ) }{d{\it \_b}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{a} \left ( {\it c1}\, \left ( \tanh \left ( \lambda 1\,{\it \_a} \right ) \right ) ^{{\it n1}}+{\it c2}\, \left ( \tanh \left ( {\frac {\lambda 2\, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{{\it n2}} \right ) }{d{\it \_a}}}}\]
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Added April 3, 2019.
Problem Chapter 5.4.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a \tanh ^n(\lambda x) w_x + b \tanh ^m(\mu x) w_y = c \tanh ^k(\nu x) w + p \tanh ^s(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Tanh[lambda*x]^n*D[w[x, y], x] + b*Tanh[mu*x]^m*D[w[x, y], y] == c*Tanh[nu*x]*w[x,y]+p*Tanh[beta*y]^s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \tanh ^{-n}(\lambda K[2]) \tanh (\nu K[2])}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \tanh ^{-n}(\lambda K[2]) \tanh (\nu K[2])}{a}dK[2]\right ) p \tanh ^{-n}(\lambda K[3]) \tanh ^s\left (\beta \left (y-\int _1^x\frac {b \tanh ^{-n}(\lambda K[1]) \tanh ^m(\mu K[1])}{a}dK[1]+\int _1^{K[3]}\frac {b \tanh ^{-n}(\lambda K[1]) \tanh ^m(\mu K[1])}{a}dK[1]\right )\right )}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \tanh ^{-n}(\lambda K[1]) \tanh ^m(\mu K[1])}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*tanh(lambda*x)^n*diff(w(x,y),x)+ b*tanh(mu*x)^m*diff(w(x,y),y) = c*tanh(nu*x)*w[x,y]+p*tanh(beta*y)^s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{a} \left ( -b\int \! \left ( {\frac {\sinh \left ( \mu \,x \right ) }{\cosh \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( x\lambda \right ) }{\cosh \left ( x\lambda \right ) }} \right ) ^{-n}\,{\rm d}x+ay \right ) } \right ) +\int ^{x}\!{\frac { \left ( \tanh \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{a} \left ( c\tanh \left ( \nu \,{\it \_b} \right ) w_{{x,y}}+ \left ( {\sinh \left ( {\frac {\beta }{a} \left ( b\int \! \left ( \tanh \left ( \mu \,{\it \_b} \right ) \right ) ^{m} \left ( \tanh \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}\,{\rm d}{\it \_b}-b\int \! \left ( {\frac {\sinh \left ( \mu \,x \right ) }{\cosh \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( x\lambda \right ) }{\cosh \left ( x\lambda \right ) }} \right ) ^{-n}\,{\rm d}x+ay \right ) } \right ) \left ( \cosh \left ( {\frac {\beta }{a} \left ( b\int \! \left ( \tanh \left ( \mu \,{\it \_b} \right ) \right ) ^{m} \left ( \tanh \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}\,{\rm d}{\it \_b}-b\int \! \left ( {\frac {\sinh \left ( \mu \,x \right ) }{\cosh \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( x\lambda \right ) }{\cosh \left ( x\lambda \right ) }} \right ) ^{-n}\,{\rm d}x+ay \right ) } \right ) \right ) ^{-1}} \right ) ^{s}p \right ) }{d{\it \_b}}\]
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Added April 3, 2019.
Problem Chapter 5.4.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a \tanh ^n(\lambda x) w_x + b \tanh ^m(\mu x) w_y = c \tanh ^k(\nu y) w + p \tanh ^s(\beta x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Tanh[lambda*x]^n*D[w[x, y], x] + b*Tanh[mu*x]^m*D[w[x, y], y] == c*Tanh[nu*y]*w[x,y]+p*Tanh[beta*x]^s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \tanh ^{-n}(\lambda K[2]) \tanh \left (\nu \left (y-\int _1^x\frac {b \tanh ^{-n}(\lambda K[1]) \tanh ^m(\mu K[1])}{a}dK[1]+\int _1^{K[2]}\frac {b \tanh ^{-n}(\lambda K[1]) \tanh ^m(\mu K[1])}{a}dK[1]\right )\right )}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \tanh ^{-n}(\lambda K[2]) \tanh \left (\nu \left (y-\int _1^x\frac {b \tanh ^{-n}(\lambda K[1]) \tanh ^m(\mu K[1])}{a}dK[1]+\int _1^{K[2]}\frac {b \tanh ^{-n}(\lambda K[1]) \tanh ^m(\mu K[1])}{a}dK[1]\right )\right )}{a}dK[2]\right ) p \tanh ^s(\beta K[3]) \tanh ^{-n}(\lambda K[3])}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \tanh ^{-n}(\lambda K[1]) \tanh ^m(\mu K[1])}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*tanh(lambda*x)^n*diff(w(x,y),x)+ b*tanh(mu*x)^m*diff(w(x,y),y) = c*tanh(nu*y)*w(x,y)+p*tanh(beta*x)^s; 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 {p}{a} \left ( {\frac {\sinh \left ( {\it \_f}\,\lambda \right ) }{\cosh \left ( {\it \_f}\,\lambda \right ) }} \right ) ^{-n} \left ( {\frac {\sinh \left ( \beta \,{\it \_f} \right ) }{\cosh \left ( \beta \,{\it \_f} \right ) }} \right ) ^{s}{{\rm e}^{-{\frac {c}{a}\int \!{\sinh \left ( {\frac {\nu }{a} \left ( b\int \! \left ( {\frac {\sinh \left ( \mu \,{\it \_f} \right ) }{\cosh \left ( \mu \,{\it \_f} \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( {\it \_f}\,\lambda \right ) }{\cosh \left ( {\it \_f}\,\lambda \right ) }} \right ) ^{-n}\,{\rm d}{\it \_f}-b\int \! \left ( {\frac {\sinh \left ( \mu \,x \right ) }{\cosh \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( x\lambda \right ) }{\cosh \left ( x\lambda \right ) }} \right ) ^{-n}\,{\rm d}x+ay \right ) } \right ) \left ( {\frac {\sinh \left ( {\it \_f}\,\lambda \right ) }{\cosh \left ( {\it \_f}\,\lambda \right ) }} \right ) ^{-n} \left ( \cosh \left ( {\frac {\nu }{a} \left ( b\int \! \left ( {\frac {\sinh \left ( \mu \,{\it \_f} \right ) }{\cosh \left ( \mu \,{\it \_f} \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( {\it \_f}\,\lambda \right ) }{\cosh \left ( {\it \_f}\,\lambda \right ) }} \right ) ^{-n}\,{\rm d}{\it \_f}-b\int \! \left ( {\frac {\sinh \left ( \mu \,x \right ) }{\cosh \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( x\lambda \right ) }{\cosh \left ( x\lambda \right ) }} \right ) ^{-n}\,{\rm d}x+ay \right ) } \right ) \right ) ^{-1}}\,{\rm d}{\it \_f}}}}}{d{\it \_f}}+{\it \_F1} \left ( {\frac {1}{a} \left ( -b\int \! \left ( {\frac {\sinh \left ( \mu \,x \right ) }{\cosh \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( x\lambda \right ) }{\cosh \left ( x\lambda \right ) }} \right ) ^{-n}\,{\rm d}x+ay \right ) } \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac { \left ( \tanh \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}c}{a}\tanh \left ( {\frac {\nu }{a} \left ( -b\int \! \left ( {\frac {\sinh \left ( \mu \,x \right ) }{\cosh \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( x\lambda \right ) }{\cosh \left ( x\lambda \right ) }} \right ) ^{-n}\,{\rm d}x+a \left ( \int \!{\frac {b \left ( \tanh \left ( \mu \,{\it \_b} \right ) \right ) ^{m} \left ( \tanh \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{a}}\,{\rm d}{\it \_b}+y \right ) \right ) } \right ) }{d{\it \_b}}}}\]
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