Added Jan 19, 2020.
Problem Chapter 9.4.5.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a w_y + b w_z = c \sinh ^n(\beta x) w + k \cosh ^m(\lambda x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ a*D[w[x,y,z],y]+b*D[w[x,y,z],z]==c*Sinh[beta*x]^n*w[x,y,z]+ k*Cosh[lambda*x]^m; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {c \sqrt {\cosh ^2(\beta x)} \text {sech}(\beta x) \sinh ^{n+1}(\beta x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};-\sinh ^2(\beta x)\right )}{\beta n+\beta }\right ) \left (\int _1^x\exp \left (-\frac {c \sqrt {\cosh ^2(\beta K[1])} \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};-\sinh ^2(\beta K[1])\right ) \text {sech}(\beta K[1]) \sinh ^{n+1}(\beta K[1])}{n \beta +\beta }\right ) k \cosh ^m(\lambda K[1])dK[1]+c_1(y-a x,z-b x)\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ a*diff(w(x,y,z),y)+ b*diff(w(x,y,z),z)=c*sinh(beta*x)^n*w(x,y,z)+ k*cosh(lambda*x)^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) = \left ( \int \!k \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{m}{{\rm e}^{-c\int \! \left ( \sinh \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x}}\,{\rm d}x+{\it \_F1} \left ( -ax+y,-bx+z \right ) \right ) {{\rm e}^{\int \! \left ( \sinh \left ( \beta \,x \right ) \right ) ^{n}c\,{\rm d}x}}\]
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Added Jan 19, 2020.
Problem Chapter 9.4.5.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a w_y + b w_z = c \tanh ^n(\beta x) w + k \coth ^m(\lambda x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ a*D[w[x,y,z],y]+b*D[w[x,y,z],z]==c*Tanh[beta*x]^n*w[x,y,z]+ k*Coth[lambda*x]^m; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {c \tanh ^{n+1}(\beta x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\tanh ^2(\beta x)\right )}{\beta n+\beta }\right ) \left (\int _1^x\exp \left (-\frac {c \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\tanh ^2(\beta K[1])\right ) \tanh ^{n+1}(\beta K[1])}{n \beta +\beta }\right ) k \coth ^m(\lambda K[1])dK[1]+c_1(y-a x,z-b x)\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ a*diff(w(x,y,z),y)+ b*diff(w(x,y,z),z)=c*tanh(beta*x)^n*w(x,y,z)+ k*coth(lambda*x)^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) = \left ( \int \!k \left ( {\frac {\cosh \left ( \lambda \,x \right ) }{\sinh \left ( \lambda \,x \right ) }} \right ) ^{m}{{\rm e}^{-c\int \! \left ( {\frac {\sinh \left ( \beta \,x \right ) }{\cosh \left ( \beta \,x \right ) }} \right ) ^{n}\,{\rm d}x}}\,{\rm d}x+{\it \_F1} \left ( -ax+y,-bx+z \right ) \right ) {{\rm e}^{\int \! \left ( {\frac {\sinh \left ( \beta \,x \right ) }{\cosh \left ( \beta \,x \right ) }} \right ) ^{n}c\,{\rm d}x}}\]
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Added Jan 19, 2020.
Problem Chapter 9.4.5.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + b \cosh ^n(\beta x) w_y + c \sinh ^k(\lambda y) w_z = a w + s \cosh ^m(\mu x) \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ b*Cosh[beta*x]^n*D[w[x,y,z],y]+c*Sinh[lambda*y]^k*D[w[x,y,z],z]==a*w[x,y,z]+ s*Cosh[mu*x]^m; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ b*cosh(beta*x)^n*diff(w(x,y,z),y)+ c*sinh(lambda*x)^k*diff(w(x,y,z),z)=a*w(x,y,z)+ k*cosh(mu*x)^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) = \left ( \int \!k \left ( \cosh \left ( \mu \,x \right ) \right ) ^{m}{{\rm e}^{-ax}}\,{\rm d}x+{\it \_F1} \left ( -\int \!b \left ( \cosh \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int \!c \left ( \sinh \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+z \right ) \right ) {{\rm e}^{ax}}\]
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Added Jan 19, 2020.
Problem Chapter 9.4.5.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a \tanh ^n(\beta x) w_y + b \coth ^k(\lambda x) w_z = c w + s \tanh ^m(\mu x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ a*Tanh[beta*x]^n*D[w[x,y,z],y]+b*Coth[lambda*x]^k*D[w[x,y,z],z]==c*w[x,y,z]+ s*Tanh[mu*x]^m; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to -\frac {s \left (e^{-2 \mu x}-1\right )^m \left (e^{-2 \mu x}+1\right )^m \left (-e^{-4 \mu x} \left (e^{2 \mu x}-1\right )^2\right )^{-m} \tanh ^m(\mu x) F_1\left (\frac {c}{2 \mu };m,-m;\frac {c}{2 \mu }+1;-e^{-2 \mu x},e^{-2 \mu x}\right )}{c}+e^{c x} c_1\left (z-\frac {b \coth ^{k+1}(\lambda x) \, _2F_1\left (1,\frac {k+1}{2};\frac {k+3}{2};\coth ^2(\lambda x)\right )}{k \lambda +\lambda },y-\frac {a \tanh ^{n+1}(\beta x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\tanh ^2(\beta x)\right )}{\beta n+\beta }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ a*tanh(beta*x)^n*diff(w(x,y,z),y)+ b*coth(lambda*x)^k*diff(w(x,y,z),z)=c*w(x,y,z)+ s*tanh(mu*x)^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) = \left ( \int \!s \left ( \tanh \left ( \mu \,x \right ) \right ) ^{m}{{\rm e}^{-xc}}\,{\rm d}x+{\it \_F1} \left ( -\int \!a \left ( \tanh \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( {\rm coth} \left (\lambda \,x\right ) \right ) ^{k}\,{\rm d}x+z \right ) \right ) {{\rm e}^{xc}}\]
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Added Jan 19, 2020.
Problem Chapter 9.4.5.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ b_1 \sinh ^{n_1}(\lambda _1 x) w_x + b_2 \cosh ^{n_2}(\lambda _2 y) w_y + b_3 \sinh ^{n_3}(\lambda _3 z) w_z = a w + c_1 \cosh ^{k_1}(\beta _1 x)+ c_2 \sinh ^{k_2}(\beta _2 y)+ c_3 \sinh ^{k_3}(\beta _3 z) \]
Mathematica ✓
ClearAll["Global`*"]; pde = b1*Sinh[lambda1*x]^n1*D[w[x,y,z],x]+ b2*Cosh[lambda2*x]^n2*D[w[x,y,z],y]+b3*Sinh[lambda3*x]^n3*D[w[x,y,z],z]==a*w[x,y,z]+ c1*Cosh[beta1*x]^k1+ c2*Sinh[beta2*x]^k2+ c3*Sinh[beta3*x]^k3; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {a \sqrt {\cosh ^2(\text {lambda1} x)} \text {sech}(\text {lambda1} x) \sinh ^{1-\text {n1}}(\text {lambda1} x) \, _2F_1\left (\frac {1}{2},\frac {1-\text {n1}}{2};\frac {3-\text {n1}}{2};-\sinh ^2(\text {lambda1} x)\right )}{\text {b1} \text {lambda1}-\text {b1} \text {lambda1} \text {n1}}\right ) \left (\int _1^x\frac {\exp \left (\frac {a \sqrt {\cosh ^2(\text {lambda1} K[3])} \, _2F_1\left (\frac {1}{2},\frac {1-\text {n1}}{2};\frac {3-\text {n1}}{2};-\sinh ^2(\text {lambda1} K[3])\right ) \text {sech}(\text {lambda1} K[3]) \sinh ^{1-\text {n1}}(\text {lambda1} K[3])}{\text {b1} \text {lambda1} \text {n1}-\text {b1} \text {lambda1}}\right ) \left (\text {c1} \cosh ^{\text {k1}}(\text {beta1} K[3])+\text {c2} \sinh ^{\text {k2}}(\text {beta2} K[3])+\text {c3} \sinh ^{\text {k3}}(\text {beta3} K[3])\right ) \sinh ^{-\text {n1}}(\text {lambda1} K[3])}{\text {b1}}dK[3]+c_1\left (y-\int _1^x\frac {\text {b2} \cosh ^{\text {n2}}(\text {lambda2} K[1]) \sinh ^{-\text {n1}}(\text {lambda1} K[1])}{\text {b1}}dK[1],z-\int _1^x\frac {\text {b3} \sinh ^{-\text {n1}}(\text {lambda1} K[2]) \sinh ^{\text {n3}}(\text {lambda3} K[2])}{\text {b1}}dK[2]\right )\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := b__1*sinh(lambda__1*x)^(n__1)*diff(w(x,y,z),x)+b__2*cosh(lambda__2*x)^(n__2)*diff(w(x,y,z),y)+ b__3*sinh(lambda__3*x)^(n__3)*diff(w(x,y,z),z)=a*w(x,y,z)+ c__1*cosh(beta__1*x)^(k__1)+ c__2*sinh(beta__2*x)^(k__2)+ c__3*sinh(beta__3*x)^(k__3); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) = \left ( \int \!{\frac { \left ( c_{1}\, \left ( \cosh \left ( \beta _{1}\,x \right ) \right ) ^{k_{1}}+c_{2}\, \left ( \sinh \left ( \beta _{2}\,x \right ) \right ) ^{k_{2}}+c_{3}\, \left ( \sinh \left ( \beta _{3}\,x \right ) \right ) ^{k_{3}} \right ) \left ( \sinh \left ( \lambda _{1}\,x \right ) \right ) ^{-n_{1}}}{b_{1}}{{\rm e}^{-{\frac {a\int \! \left ( \sinh \left ( \lambda _{1}\,x \right ) \right ) ^{-n_{1}}\,{\rm d}x}{b_{1}}}}}}\,{\rm d}x+{\it \_F1} \left ( {\frac {yb_{1}-b_{2}\,\int \! \left ( \cosh \left ( \lambda _{2}\,x \right ) \right ) ^{n_{2}} \left ( \sinh \left ( \lambda _{1}\,x \right ) \right ) ^{-n_{1}}\,{\rm d}x}{b_{1}}},{\frac {-b_{3}\,\int \! \left ( \sinh \left ( \lambda _{3}\,x \right ) \right ) ^{n_{3}} \left ( \sinh \left ( \lambda _{1}\,x \right ) \right ) ^{-n_{1}}\,{\rm d}x+b_{1}\,z}{b_{1}}} \right ) \right ) {{\rm e}^{\int \!{\frac {a \left ( \sinh \left ( \lambda _{1}\,x \right ) \right ) ^{-n_{1}}}{b_{1}}}\,{\rm d}x}}\]
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