Added Oct 10, 2019.
Problem Chapter 8.4.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a w_y + b w_z = c \tanh ^n(\beta x) w \]
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]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {c \tanh ^{n+1}(\beta x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\tanh ^2(\beta x)\right )}{\beta n+\beta }\right ) c_1(y-a x,z-b x)\right \}\right \}\]
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); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -ax+y,-bx+z \right ) {{\rm e}^{\int \! \left ( \tanh \left ( \beta \,x \right ) \right ) ^{n}c\,{\rm d}x}}\]
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Added Oct 10, 2019.
Problem Chapter 8.4.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b w_y + c \tanh (\lambda x) w_z = \left ( k \tanh (\beta x)+s \tanh (\gamma z) \right ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*Tanh[lambda*x]*D[w[x,y,z],z]== (k*Tanh[beta*x]+s*Tanh[gamma*z])*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},z-\frac {c \log (\cosh (\lambda x))}{a \lambda }\right ) \exp \left (\int _1^x\frac {k \tanh (\beta K[1])+s \tanh \left (\frac {\gamma (a \lambda z-c \log (\cosh (\lambda x))+c \log (\cosh (\lambda K[1])))}{a \lambda }\right )}{a}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x, y,z), x) + b*diff(w(x, y,z), y) + c*tanh(lambda*x)*diff(w(x,y,z),z)= (k*tanh(beta*x)+s*tanh(gamma*z))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}},1/2\,{\frac {2\,za\lambda +c\ln \left ( \tanh \left ( \lambda \,x \right ) -1 \right ) +c\ln \left ( \tanh \left ( \lambda \,x \right ) +1 \right ) }{a\lambda }} \right ) {{\rm e}^{-\int ^{x}\!{\frac {1}{a} \left ( -k\tanh \left ( \beta \,{\it \_a} \right ) +s\tanh \left ( 1/16\,{\frac {-2\,za\lambda -c\ln \left ( \tanh \left ( \lambda \,x \right ) -1 \right ) -c\ln \left ( \tanh \left ( \lambda \,x \right ) +1 \right ) +c\ln \left ( \tanh \left ( {\it \_a}\,\lambda \right ) -1 \right ) +c\ln \left ( \tanh \left ( {\it \_a}\,\lambda \right ) +1 \right ) }{a\lambda }} \right ) \right ) }{d{\it \_a}}}}\]
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Added Oct 10, 2019.
Problem Chapter 8.4.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a \tanh ^n(\beta x) w_y + b \tanh ^k(\lambda x) w_z = c \tanh ^m(\gamma x) w \]
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*Tanh[gamma*x]^m *w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {c \tanh ^{m+1}(\gamma x) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};\tanh ^2(\gamma x)\right )}{\gamma m+\gamma }\right ) c_1\left (y-\frac {a \tanh ^{n+1}(\beta x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\tanh ^2(\beta x)\right )}{\beta n+\beta },z-\frac {b \tanh ^{k+1}(\lambda x) \, _2F_1\left (1,\frac {k+1}{2};\frac {k+3}{2};\tanh ^2(\lambda x)\right )}{k \lambda +\lambda }\right )\right \}\right \}\]
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*tanh(gamma*x)^m *w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!a \left ( \tanh \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( \tanh \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+z \right ) {{\rm e}^{\int \!c \left ( \tanh \left ( x/8 \right ) \right ) ^{m}\,{\rm d}x}}\]
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Added Oct 10, 2019.
Problem Chapter 8.4.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b \tanh (\beta y) w_y + c \tanh (\lambda x) w_z = k \tanh (\gamma z) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Tanh[beta*y]*D[w[x, y,z], y] + c*Tanh[lambda*x]*D[w[x,y,z],z]== k*Tanh[gamma*z] *w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := a*diff(w(x, y,z), x) + b*tanh(beta*y)*diff(w(x, y,z), y) + c*tanh(lambda*x)*diff(w(x,y,z),z)= k*tanh(gamma*z) *w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( 1/2\,{\frac {2\,bx\beta +a\ln \left ( \tanh \left ( \beta \,y \right ) -1 \right ) +a\ln \left ( \tanh \left ( \beta \,y \right ) +1 \right ) -2\,a\ln \left ( \tanh \left ( \beta \,y \right ) \right ) }{b\beta }},{\frac {za\lambda -c\ln \left ( \cosh \left ( \lambda \,x \right ) \right ) }{a\lambda }} \right ) {{\rm e}^{\int ^{y}\!{\frac {k}{b\tanh \left ( \beta \,{\it \_a} \right ) }\tanh \left ( 1/8\,{\frac {1}{a\lambda } \left ( za\lambda -c\ln \left ( \cosh \left ( \lambda \,x \right ) \right ) +c\ln \left ( \cosh \left ( 1/2\,{\frac {\lambda \, \left ( -2\,bx\beta -a\ln \left ( \tanh \left ( \beta \,y \right ) -1 \right ) -a\ln \left ( \tanh \left ( \beta \,y \right ) +1 \right ) +2\,a\ln \left ( \tanh \left ( \beta \,y \right ) \right ) +a\ln \left ( \tanh \left ( \beta \,{\it \_a} \right ) -1 \right ) +a\ln \left ( \tanh \left ( \beta \,{\it \_a} \right ) +1 \right ) -2\,a\ln \left ( \tanh \left ( \beta \,{\it \_a} \right ) \right ) \right ) }{b\beta }} \right ) \right ) \right ) } \right ) }{d{\it \_a}}}}\]
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Added Oct 10, 2019.
Problem Chapter 8.4.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b \tanh (\beta y) w_y + c \tanh (\gamma z) w_z = k \tanh (\lambda x) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Tanh[beta*y]*D[w[x, y,z], y] + c*Tanh[gamma*z]*D[w[x,y,z],z]== k*Tanh[lambda*x] *w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \cosh ^{\frac {k}{a \lambda }}(\lambda x) c_1\left (\frac {1}{2} \left (\frac {\log (\sinh (\beta y))}{\beta }-\frac {b x}{a}\right ),\frac {b \log \left (\sinh ^2(\gamma z)\right )}{\gamma }-\frac {2 c \log (\sinh (\beta y))}{\beta }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x, y,z), x) + b*tanh(beta*y)*diff(w(x, y,z), y) + c*tanh(gamma*z)*diff(w(x,y,z),z)= k*tanh(lambda*x) *w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( 1/2\,{\frac {2\,bx\beta +a\ln \left ( \tanh \left ( \beta \,y \right ) -1 \right ) +a\ln \left ( \tanh \left ( \beta \,y \right ) +1 \right ) -2\,a\ln \left ( \tanh \left ( \beta \,y \right ) \right ) }{b\beta }},1/2\,{\frac {1}{\beta \,c} \left ( 8\,\ln \left ( {\frac { \left ( \RootOf \left ( 8\,\arctanh \left ( {\it \_Z} \right ) +z \right ) \right ) ^{2}}{ \left ( \RootOf \left ( 8\,\arctanh \left ( {\it \_Z} \right ) +z \right ) \right ) ^{2}-1}} \right ) b\beta +2\,\beta \,cy+c\ln \left ( -4\, \left ( {{\rm e}^{2\,\beta \,y}}-1 \right ) ^{-2} \right ) \right ) } \right ) \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{{\frac {k}{a\lambda }}}\]
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Added Oct 10, 2019.
Problem Chapter 8.4.3.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a_1 \tanh ^{n_1}(\lambda _1 x) w_x + b_1 \tanh ^{m_1}(\beta _1 y) w_y + c_1 \tanh ^{k_1}(\gamma _1 z) w_z = \left ( a_2 \tanh ^{n_2}(\lambda _2 x) w_x + b_2 \tanh ^{m_2}(\beta _2 y) w_y + c_2 \tanh ^{k_2}(\gamma _2 z) \right ) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a1*Tanh[lambda1*x]^n1*D[w[x, y,z], x] + b1*Tanh[beta1*y]^m1*D[w[x, y,z], y] + c1*Tanh[gamma1*x]^k1*D[w[x, y,z], z]== (a2*Tanh[lambda2*x]^n2+b2*Tanh[beta2*y]^m2+c2*Tanh[gamma2*x]^k2) *w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := a1*tanh(lambda1*x)^n1*diff(w(x, y,z), x) + b1*tanh(beta1*y)^m1*diff(w(x, y,z), y) + c1*tanh(gamma1*x)^k1*diff(w(x,y,z),z)= ( a2*tanh(lambda2*x)^n2+b2*tanh(beta2*y)^m2+c2*tanh(gamma2*x)^k2) *w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \! \left ( \tanh \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( \tanh \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}{\it a1}}{{\it b1}}}\,{\rm d}y,{\frac {1}{{\it a1}} \left ( z{\it a1}-{\it c1}\,\int \! \left ( {\frac {\sinh \left ( \gamma 1\,x \right ) }{\cosh \left ( \gamma 1\,x \right ) }} \right ) ^{{\it k1}} \left ( {\frac {\sinh \left ( \lambda 1\,x \right ) }{\cosh \left ( \lambda 1\,x \right ) }} \right ) ^{-{\it n1}}\,{\rm d}x \right ) } \right ) {{\rm e}^{\int ^{x}\!{\frac { \left ( \tanh \left ( \lambda 1\,{\it \_f} \right ) \right ) ^{-{\it n1}}}{{\it a1}} \left ( {\it b2}\, \left ( \tanh \left ( \beta 2\,\RootOf \left ( \int \! \left ( \tanh \left ( \lambda 1\,{\it \_f} \right ) \right ) ^{-{\it n1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \tanh \left ( \beta 1\,{\it \_a} \right ) \right ) ^{-{\it m1}}{\it a1}}{{\it b1}}}{d{\it \_a}}-\int \! \left ( \tanh \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( \tanh \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}{\it a1}}{{\it b1}}}\,{\rm d}y \right ) \right ) \right ) ^{{\it m2}}+{\it a2}\, \left ( \tanh \left ( \lambda 2\,{\it \_f} \right ) \right ) ^{{\it n2}}+{\it c2}\, \left ( \tanh \left ( \gamma 2\,{\it \_f} \right ) \right ) ^{{\it k2}} \right ) }{d{\it \_f}}}}\]
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