Added Jan 19, 2020.
Problem Chapter 9.4.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a w_y + b w_z = c \tanh ^n(\beta x) w + k \tanh ^m(\lambda x) \]
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*Tanh[lambda*x]^m; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {c \tanh ^{n+1}(\beta x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\tanh ^2(\beta x)\right )}{\beta n+\beta }\right ) \left (\int _1^x\exp \left (-\frac {c \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\tanh ^2(\beta K[1])\right ) \tanh ^{n+1}(\beta K[1])}{n \beta +\beta }\right ) k \tanh ^m(\lambda K[1])dK[1]+c_1(y-a x,z-b x)\right )\right \}\right \}\]
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*tanh(lambda*x)^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \left (\int k \left (\tanh ^{m}\left (\lambda x \right )\right ) {\mathrm e}^{-c \left (\int \left (\tanh ^{n}\left (\beta x \right )\right )d x \right )}d x +\mathit {\_F1} \left (-a x +y , -x b +z \right )\right ) {\mathrm e}^{\int c \left (\tanh ^{n}\left (\beta x \right )\right )d x}\]
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Added Jan 19, 2020.
Problem Chapter 9.4.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b w_y + c \tanh (\beta z) w_z = \left (p \tanh (\lambda x) + q \right ) w + k \tanh (\gamma x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x,y,z],x]+ b*D[w[x,y,z],y]+c*Tanh[beta*z]*D[w[x,y,z],z]==(p*Tanh[lambda*x]+q)*w[x,y,z]+ k*Tanh[gamma*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {q x}{a}} \cosh ^{\frac {p}{a \lambda }}(\lambda x) \left (\int _1^x\frac {e^{-\frac {q K[1]}{a}} k \cosh ^{-\frac {p}{a \lambda }}(\lambda K[1]) \tanh (\gamma K[1])}{a}dK[1]+c_1\left (y-\frac {b x}{a},\frac {\log (\sinh (\beta z))}{\beta }-\frac {c x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ c*tanh(beta*z)*diff(w(x,y,z),z)=(p*tanh(lambda*x)+q)*w(x,y,z)+ k*tanh(gamma*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \left (-\left (\int _{}^{z}\frac {k \left (\tanh \left (\frac {\left (-2 \beta c x +a \ln \left (\tanh \left (\mathit {\_a} \beta \right )-1\right )+a \ln \left (\tanh \left (\mathit {\_a} \beta \right )+1\right )-a \ln \left (\tanh \left (\beta z \right )-1\right )-a \ln \left (\tanh \left (\beta z \right )+1\right )-2 a \ln \left (\tanh \left (\mathit {\_a} \beta \right )\right )+2 a \ln \left (\tanh \left (\beta z \right )\right )\right ) \lambda }{2 \beta c}\right )-1\right )^{\frac {p -q}{2 a \lambda }} \left (\tanh \left (\frac {\left (-2 \beta c x +a \ln \left (\tanh \left (\mathit {\_a} \beta \right )-1\right )+a \ln \left (\tanh \left (\mathit {\_a} \beta \right )+1\right )-a \ln \left (\tanh \left (\beta z \right )-1\right )-a \ln \left (\tanh \left (\beta z \right )+1\right )-2 a \ln \left (\tanh \left (\mathit {\_a} \beta \right )\right )+2 a \ln \left (\tanh \left (\beta z \right )\right )\right ) \lambda }{2 \beta c}\right )+1\right )^{\frac {p +q}{2 a \lambda }} \tanh \left (\frac {\left (-2 \beta c x +a \ln \left (\tanh \left (\mathit {\_a} \beta \right )-1\right )+a \ln \left (\tanh \left (\mathit {\_a} \beta \right )+1\right )-a \ln \left (\tanh \left (\beta z \right )-1\right )-a \ln \left (\tanh \left (\beta z \right )+1\right )-2 a \ln \left (\tanh \left (\mathit {\_a} \beta \right )\right )+2 a \ln \left (\tanh \left (\beta z \right )\right )\right ) \gamma }{2 \beta c}\right )}{c \tanh \left (\mathit {\_a} \beta \right )}d\mathit {\_a} \right )+\mathit {\_F1} \left (\frac {2 \beta c x +a \ln \left (\tanh \left (\beta z \right )-1\right )+a \ln \left (\tanh \left (\beta z \right )+1\right )-2 a \ln \left (\tanh \left (\beta z \right )\right )}{2 \beta c}, \frac {2 \beta c y +b \ln \left (\tanh \left (\beta z \right )-1\right )+b \ln \left (\tanh \left (\beta z \right )+1\right )-2 b \ln \left (\tanh \left (\beta z \right )\right )}{2 \beta c}\right )\right ) \left (-\tanh \left (\lambda x \right )-1\right )^{-\frac {p -q}{2 a \lambda }} \left (-\tanh \left (\lambda x \right )+1\right )^{-\frac {p +q}{2 a \lambda }}\]
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Added Jan 19, 2020.
Problem Chapter 9.4.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a \tanh ^n(\beta x) w_y + b \tanh ^k(\lambda x) w_z = c w + s \tanh ^m(\mu x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ a*Tanh[beta*x]^n*D[w[x,y,z],y]+b*Tanh[lambda*x]^k*D[w[x,y,z],z]==c*w[x,y,z]+ k*Tanh[mu*x]^m; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to -\frac {k \left (e^{-2 \mu x}-1\right )^m \left (e^{-2 \mu x}+1\right )^m \left (-e^{-4 \mu x} \left (e^{2 \mu x}-1\right )^2\right )^{-m} \tanh ^m(\mu x) F_1\left (\frac {c}{2 \mu };m,-m;\frac {c}{2 \mu }+1;-e^{-2 \mu x},e^{-2 \mu x}\right )}{c}+e^{c x} c_1\left (y-\frac {a \tanh ^{n+1}(\beta x) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},\tanh ^2(\beta x)\right )}{\beta n+\beta },z-\frac {b \tanh ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {k+1}{2},\frac {k+3}{2},\tanh ^2(\lambda x)\right )}{k \lambda +\lambda }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+ a*tanh(beta*x)^n*diff(w(x,y,z),y)+ b*tanh(lambda*x)^k*diff(w(x,y,z),z)=c*w(x,y,z)+ k*tanh(mu*x)^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \left (\int k \left (\tanh ^{m}\left (\mu x \right )\right ) {\mathrm e}^{-c x}d x +\mathit {\_F1} \left (y -\left (\int a \left (\tanh ^{n}\left (\beta x \right )\right )d x \right ), z -\left (\int b \left (\tanh ^{k}\left (\lambda x \right )\right )d x \right )\right )\right ) {\mathrm e}^{c x}\]
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Added Jan 19, 2020.
Problem Chapter 9.4.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + b \tanh ^n(\beta x) w_y + c \tanh ^k(\lambda y) w_z = a w + s \tanh ^m(\mu x) \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x,y,z],x]+ b*Tanh[beta*x]^n*D[w[x,y,z],y]+c*Tanh[lambda*y]^k*D[w[x,y,z],z]==a*w[x,y,z]+ s*Tanh[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*tanh(beta*x)^n*diff(w(x,y,z),y)+ c*tanh(lambda*y)^k*diff(w(x,y,z),z)=a*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'));
\[w \left (x , y , z\right ) = \left (\int s \left (\tanh ^{m}\left (\mu x \right )\right ) {\mathrm e}^{-a x}d x +\mathit {\_F1} \left (y -\left (\int b \left (\tanh ^{n}\left (\beta x \right )\right )d x \right ), z -\left (\int _{}^{x}c \left (\frac {\sinh \left (\left (b \left (\int \left (\tanh ^{n}\left (\mathit {\_b} \beta \right )\right )d \mathit {\_b} \right )+y -\left (\int b \left (\tanh ^{n}\left (\beta x \right )\right )d x \right )\right ) \lambda \right )}{\cosh \left (\left (b \left (\int \left (\tanh ^{n}\left (\mathit {\_b} \beta \right )\right )d \mathit {\_b} \right )+y -\left (\int b \left (\tanh ^{n}\left (\beta x \right )\right )d x \right )\right ) \lambda \right )}\right )^{k}d\mathit {\_b} \right )\right )\right ) {\mathrm e}^{a x}\]
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Added Jan 19, 2020.
Problem Chapter 9.4.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ b_1 \tanh ^{n_1}(\lambda _1 x) w_x + b_2 \tanh ^{n_2}(\lambda _2 y) w_y + b_3 \tanh ^{n_3}(\lambda _3 z) w_z = a w + c_1 \tanh ^{k_1}(\beta _1 x)+ c_2 \tanh ^{k_2}(\beta _2 y)+ c_3 \tanh ^{k_3}(\beta _3 z) \]
Mathematica ✓
ClearAll["Global`*"]; pde = b1*Tanh[lambda1*x]^n1*D[w[x,y,z],x]+ b2*Tanh[lambda2*x]^n2*D[w[x,y,z],y]+b3*Tanh[lambda3*x]^n3*D[w[x,y,z],z]==a*w[x,y,z]+ c1*Tanh[beta1*x]^k1+ c2*Tanh[beta2*x]^k2+ c3*Tanh[beta3*x]^k3; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {a \tanh ^{1-\text {n1}}(\text {lambda1} x) \, _2F_1\left (1,\frac {1}{2}-\frac {\text {n1}}{2};\frac {3}{2}-\frac {\text {n1}}{2};\tanh ^2(\text {lambda1} x)\right )}{\text {b1} \text {lambda1}-\text {b1} \text {lambda1} \text {n1}}\right ) \left (\int _1^x\frac {\exp \left (\frac {a \, _2F_1\left (1,\frac {1}{2}-\frac {\text {n1}}{2};\frac {3}{2}-\frac {\text {n1}}{2};\tanh ^2(\text {lambda1} K[3])\right ) \tanh ^{1-\text {n1}}(\text {lambda1} K[3])}{\text {b1} \text {lambda1} \text {n1}-\text {b1} \text {lambda1}}\right ) \left (\text {c1} \tanh ^{\text {k1}}(\text {beta1} K[3])+\text {c2} \tanh ^{\text {k2}}(\text {beta2} K[3])+\text {c3} \tanh ^{\text {k3}}(\text {beta3} K[3])\right ) \tanh ^{-\text {n1}}(\text {lambda1} K[3])}{\text {b1}}dK[3]+c_1\left (y-\int _1^x\frac {\text {b2} \tanh ^{-\text {n1}}(\text {lambda1} K[1]) \tanh ^{\text {n2}}(\text {lambda2} K[1])}{\text {b1}}dK[1],z-\int _1^x\frac {\text {b3} \tanh ^{-\text {n1}}(\text {lambda1} K[2]) \tanh ^{\text {n3}}(\text {lambda3} K[2])}{\text {b1}}dK[2]\right )\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := b__1*tanh(lambda__1*x)^(n__1)*diff(w(x,y,z),x)+b__2*tanh(lambda__2*x)^(n__2)*diff(w(x,y,z),y)+ b__3*tanh(lambda__3*x)^(n__3)*diff(w(x,y,z),z)=a*w(x,y,z)+ c__1*tanh(beta__1*x)^(k__1)+ c__2*tanh(beta__2*x)^(k__2)+ c__3*tanh(beta__3*x)^(k__3); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \left (\int \frac {\left (c_{1} \left (\tanh ^{k_{1}}\left (\beta _{1} x \right )\right )+c_{2} \left (\tanh ^{k_{2}}\left (\beta _{2} x \right )\right )+c_{3} \left (\tanh ^{k_{3}}\left (\beta _{3} x \right )\right )\right ) \left (\tanh ^{-n_{1}}\left (x \lambda _{1} \right )\right ) {\mathrm e}^{-\frac {a \left (\int \left (\tanh ^{-n_{1}}\left (x \lambda _{1} \right )\right )d x \right )}{b_{1}}}}{b_{1}}d x +\mathit {\_F1} \left (\frac {b_{1} y -b_{2} \left (\int \left (\frac {\sinh \left (x \lambda _{1} \right )}{\cosh \left (x \lambda _{1} \right )}\right )^{-n_{1}} \left (\frac {\sinh \left (x \lambda _{2} \right )}{\cosh \left (x \lambda _{2} \right )}\right )^{n_{2}}d x \right )}{b_{1}}, \frac {b_{1} z -b_{3} \left (\int \left (\frac {\sinh \left (x \lambda _{1} \right )}{\cosh \left (x \lambda _{1} \right )}\right )^{-n_{1}} \left (\frac {\sinh \left (x \lambda _{3} \right )}{\cosh \left (x \lambda _{3} \right )}\right )^{n_{3}}d x \right )}{b_{1}}\right )\right ) {\mathrm e}^{\int \frac {a \left (\tanh ^{-n_{1}}\left (x \lambda _{1} \right )\right )}{b_{1}}d x}\]
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