Added Oct 10, 2019.
Problem Chapter 8.4.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a w_y + b w_z = c \coth ^n(\beta x) w \]
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*Coth[beta*x]^n*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {c \coth ^{n+1}(\beta x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\coth ^2(\beta x)\right )}{\beta n+\beta }\right ) c_1(y-a x,z-b x)\right \}\right \}\]
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*coth(beta*x)^n*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -ax+y,-xb+z \right ) {{\rm e}^{\int \!c \left ( {\rm coth} \left (\beta \,x\right ) \right ) ^{n}\,{\rm d}x}}\]
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Added Oct 10, 2019.
Problem Chapter 8.4.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b w_y + c \coth (\lambda x) w_z = \left ( k \coth (\beta x)+s \coth (\gamma z) \right ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*Coth[lambda*x]*D[w[x,y,z],z]== (k*Coth[beta*x]+s*Coth[gamma*z])*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},z-\frac {c \log (\sinh (\lambda x))}{a \lambda }\right ) \exp \left (\int _1^x\frac {k \coth (\beta K[1])+s \coth \left (\frac {\gamma (a \lambda z-c \log (\sinh (\lambda x))+c \log (\sinh (\lambda K[1])))}{a \lambda }\right )}{a}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x, y,z), x) + b*diff(w(x, y,z), y) + c*coth(lambda*x)*diff(w(x,y,z),z)= (k*coth(beta*x)+s*coth(gamma*z))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ay-xb}{a}},{\frac {2\,za\lambda +c\ln \left ( {\rm coth} \left (x\lambda \right )-1 \right ) +c\ln \left ( {\rm coth} \left (x\lambda \right )+1 \right ) }{2\,a\lambda }} \right ) {{\rm e}^{-\int ^{x}\!{\frac {1}{a} \left ( -k{\rm coth} \left (\beta \,{\it \_a}\right )+s{\rm coth} \left ({\frac {\gamma \, \left ( -2\,za\lambda -c\ln \left ( {\rm coth} \left (x\lambda \right )-1 \right ) -c\ln \left ( {\rm coth} \left (x\lambda \right )+1 \right ) +\ln \left ( {\rm coth} \left ({\it \_a}\,\lambda \right )-1 \right ) c+\ln \left ( {\rm coth} \left ({\it \_a}\,\lambda \right )+1 \right ) c \right ) }{2\,a\lambda }}\right ) \right ) }{d{\it \_a}}}}\]
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Added Oct 10, 2019.
Problem Chapter 8.4.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a \coth ^n(\beta x) w_y + b \coth ^k(\lambda x) w_z = c \coth ^m(\gamma x) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Coth[beta*x]^n*D[w[x, y,z], y] + b*Coth[lambda*x]^k*D[w[x,y,z],z]== c*Coth[gamma*x]^m *w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {c \coth ^{m+1}(\gamma x) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};\coth ^2(\gamma x)\right )}{\gamma m+\gamma }\right ) c_1\left (z-\frac {b \coth ^{k+1}(\lambda x) \, _2F_1\left (1,\frac {k+1}{2};\frac {k+3}{2};\coth ^2(\lambda x)\right )}{k \lambda +\lambda },y-\frac {a \coth ^{n+1}(\beta x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\coth ^2(\beta x)\right )}{\beta n+\beta }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x, y,z), x) + a*coth(beta*x)^n*diff(w(x, y,z), y) + b*coth(lambda*x)^k*diff(w(x,y,z),z)= c*coth(gamma*x)^m *w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!a \left ( {\rm coth} \left (\beta \,x\right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( {\rm coth} \left (x\lambda \right ) \right ) ^{k}\,{\rm d}x+z \right ) {{\rm e}^{\int \!c \left ( {\rm coth} \left (x\gamma \right ) \right ) ^{m}\,{\rm d}x}}\]
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Added Oct 10, 2019.
Problem Chapter 8.4.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b \coth (\beta y) w_y + c \coth (\lambda x) w_z = k \coth (\gamma z) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Coth[beta*y]*D[w[x, y,z], y] + c*Coth[lambda*x]*D[w[x,y,z],z]== k*Coth[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*coth(beta*y)*diff(w(x, y,z), y) + c*coth(lambda*x)*diff(w(x,y,z),z)= k*coth(gamma*z) *w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {1}{2\,\beta \,b} \left ( -2\,b\beta \,x+a\ln \left ( {\frac { \left ( \RootOf \left ( \beta \,y-{\rm arccoth} \left ({\it \_Z}-1\right ) \right ) -1 \right ) ^{2}}{\RootOf \left ( \beta \,y-{\rm arccoth} \left ({\it \_Z}-1\right ) \right ) -2}} \right ) -\ln \left ( \RootOf \left ( \beta \,y-{\rm arccoth} \left ({\it \_Z}-1\right ) \right ) \right ) a \right ) },{\frac {2\,za\lambda +c\ln \left ( {\rm coth} \left (x\lambda \right )-1 \right ) +c\ln \left ( {\rm coth} \left (x\lambda \right )+1 \right ) }{2\,a\lambda }} \right ) {{\rm e}^{-\int ^{x}\!{\frac {k}{a}{\rm coth} \left ({\frac {\gamma \, \left ( -2\,za\lambda -c\ln \left ( {\rm coth} \left (x\lambda \right )-1 \right ) -c\ln \left ( {\rm coth} \left (x\lambda \right )+1 \right ) +\ln \left ( {\rm coth} \left ({\it \_a}\,\lambda \right )-1 \right ) c+\ln \left ( {\rm coth} \left ({\it \_a}\,\lambda \right )+1 \right ) c \right ) }{2\,a\lambda }}\right )}{d{\it \_a}}}}\]
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Added Oct 10, 2019.
Problem Chapter 8.4.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b \coth (\beta y) w_y + c \coth (\gamma z) w_z = k \coth (\lambda x) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Coth[beta*y]*D[w[x, y,z], y] + c*Coth[gamma*z]*D[w[x,y,z],z]== k*Coth[lambda*x] *w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {k (\log (-\tanh (\lambda x))+\log (\cosh (\lambda x)))}{a \lambda }} c_1\left (\frac {a \log (\text {sech}(\beta y))+b \beta x}{2 a \beta },\frac {2 c \log (\text {sech}(\beta y))}{\beta }-\frac {b \log \left (\text {sech}^2(\gamma z)\right )}{\gamma }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x, y,z), x) + b*coth(beta*y)*diff(w(x, y,z), y) + c*coth(gamma*z)*diff(w(x,y,z),z)= k*coth(lambda*x) *w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {1}{2\,\beta \,b} \left ( -2\,b\beta \,x+a\ln \left ( {\frac { \left ( \RootOf \left ( \beta \,y-{\rm arccoth} \left ({\it \_Z}-1\right ) \right ) -1 \right ) ^{2}}{\RootOf \left ( \beta \,y-{\rm arccoth} \left ({\it \_Z}-1\right ) \right ) -2}} \right ) -\ln \left ( \RootOf \left ( \beta \,y-{\rm arccoth} \left ({\it \_Z}-1\right ) \right ) \right ) a \right ) },{\frac {1}{2\,c\gamma } \left ( -2\,c\gamma \,x+a\ln \left ( {\frac { \left ( \RootOf \left ( \gamma \,z-{\rm arccoth} \left ({\it \_Z}-1\right ) \right ) -1 \right ) ^{2}}{\RootOf \left ( \gamma \,z-{\rm arccoth} \left ({\it \_Z}-1\right ) \right ) -2}} \right ) -\ln \left ( \RootOf \left ( \gamma \,z-{\rm arccoth} \left ({\it \_Z}-1\right ) \right ) \right ) a \right ) } \right ) \left ( {\rm coth} \left (x\lambda \right )-1 \right ) ^{-{\frac {k}{2\,a\lambda }}} \left ( {\rm coth} \left (x\lambda \right )+1 \right ) ^{-{\frac {k}{2\,a\lambda }}}\]
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Added Oct 10, 2019.
Problem Chapter 8.4.4.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a_1 \coth ^{n_1}(\lambda _1 x) w_x + b_1 \coth ^{m_1}(\beta _1 y) w_y + c_1 \coth ^{k_1}(\gamma _1 z) w_z = \left ( a_2 \coth ^{n_2}(\lambda _2 x) w_x + b_2 \coth ^{m_2}(\beta _2 y) w_y + c_2 \coth ^{k_2}(\gamma _2 z) \right ) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a1*Coth[lambda1*x]^n1*D[w[x, y,z], x] + b1*Coth[beta1*y]^m1*D[w[x, y,z], y] + c1*Coth[gamma1*x]^k1*D[w[x, y,z], z]== (a2*Coth[lambda2*x]^n2+b2*Coth[beta2*y]^m2+c2*Coth[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*coth(lambda1*x)^n1*diff(w(x, y,z), x) + b1*coth(beta1*y)^m1*diff(w(x, y,z), y) + c1*coth(gamma1*x)^k1*diff(w(x,y,z),z)= ( a2*coth(lambda2*x)^n2+b2*coth(beta2*y)^m2+c2*coth(gamma2*x)^k2) *w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \! \left ( {\rm coth} \left (\lambda 1\,x\right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( {\rm coth} \left (\beta 1\,y\right ) \right ) ^{-{\it m1}}{\it a1}}{{\it b1}}}\,{\rm d}y,{\frac {1}{{\it a1}} \left ( z{\it a1}-{\it c1}\,\int \! \left ( {\frac {\cosh \left ( \lambda 1\,x \right ) }{\sinh \left ( \lambda 1\,x \right ) }} \right ) ^{-{\it n1}} \left ( {\frac {\cosh \left ( \gamma 1\,x \right ) }{\sinh \left ( \gamma 1\,x \right ) }} \right ) ^{{\it k1}}\,{\rm d}x \right ) } \right ) {{\rm e}^{\int ^{x}\!{\frac { \left ( {\rm coth} \left (\lambda 1\,{\it \_f}\right ) \right ) ^{-{\it n1}}}{{\it a1}} \left ( {\it a2}\, \left ( {\rm coth} \left (\lambda 2\,{\it \_f}\right ) \right ) ^{{\it n2}}+{\it c2}\, \left ( {\rm coth} \left (\gamma 2\,{\it \_f}\right ) \right ) ^{{\it k2}}+{\it b2}\, \left ( {\rm coth} \left (\beta 2\,\RootOf \left ( \int \! \left ( {\rm coth} \left (\lambda 1\,{\it \_f}\right ) \right ) ^{-{\it n1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( {\rm coth} \left (\beta 1\,{\it \_a}\right ) \right ) ^{-{\it m1}}{\it a1}}{{\it b1}}}{d{\it \_a}}-\int \! \left ( {\rm coth} \left (\lambda 1\,x\right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( {\rm coth} \left (\beta 1\,y\right ) \right ) ^{-{\it m1}}{\it a1}}{{\it b1}}}\,{\rm d}y \right ) \right ) \right ) ^{{\it m2}} \right ) }{d{\it \_f}}}}\]
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