Added Oct 17, 2019.
Problem Chapter 8.5.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a w_y + b w_z = c \ln ^n(\beta x) w \]
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*Log[beta*x]^n*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x) \exp \left (\frac {c (-\log (\beta x))^{-n} \log ^n(\beta x) \text {Gamma}(n+1,-\log (\beta x))}{\beta }\right )\right \}\right \}\]
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*ln(beta*x)^n*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -ax+y,-xb+z \right ) {{\rm e}^{\int \!c \left ( \ln \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x}}\]
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Added Oct 17, 2019.
Problem Chapter 8.5.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b w_y + c \ln ^n(\beta x) w_z = s \ln ^m(\lambda y) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*Log[beta*x]^n*D[w[x,y,z],z]== s*Log[lambda*y]^m*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {s (-\log (\lambda y))^{-m} \log ^m(\lambda y) \text {Gamma}(m+1,-\log (\lambda y))}{b \lambda }\right ) c_1\left (y-\frac {b x}{a},z-\frac {c (-\log (\beta x))^{-n} \log ^n(\beta x) \text {Gamma}(n+1,-\log (\beta x))}{a \beta }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+b*diff(w(x,y,z),y)+ c*ln(beta*x)^n*diff(w(x,y,z),z)= s*ln(lambda*y)^m*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ay-xb}{a}},-\int \!{\frac {c \left ( \ln \left ( \beta \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x+z \right ) {{\rm e}^{\int ^{x}\!{\frac {s}{a} \left ( \ln \left ( {\frac {\lambda \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{m}}{d{\it \_a}}}}\]
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Added Oct 17, 2019.
Problem Chapter 8.5.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a \ln ^n(\beta x) w_y + b \ln ^k(\lambda x) w_z = c \ln ^m(\gamma x) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Log[beta*x]^n*D[w[x, y,z], y] + b*Log[lambda*x]^k*D[w[x,y,z],z]== c*Log[gamma*x]^m*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {c (-\log (\gamma x))^{-m} \log ^m(\gamma x) \text {Gamma}(m+1,-\log (\gamma x))}{\gamma }\right ) c_1\left (y-\frac {a (-\log (\beta x))^{-n} \log ^n(\beta x) \text {Gamma}(n+1,-\log (\beta x))}{\beta },z-\int _1^xb \log ^k(\lambda K[1])dK[1]\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+a*ln(beta*x)^n*diff(w(x,y,z),y)+ b*ln(lambda*x)^k*diff(w(x,y,z),z)= c*ln(gamma*x)^m*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!a \left ( \ln \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( \ln \left ( x\lambda \right ) \right ) ^{k}\,{\rm d}x+z \right ) {{\rm e}^{\int \!c \left ( \ln \left ( x\gamma \right ) \right ) ^{m}\,{\rm d}x}}\]
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Added Oct 17, 2019.
Problem Chapter 8.5.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a \ln ^n(\beta x) w_y + b \ln ^k(\lambda y) w_z = c \ln ^m(\gamma x) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Log[beta*x]^n*D[w[x, y,z], y] + b*Log[lambda*y]^k*D[w[x,y,z],z]== c*Log[gamma*x]^m*w[x,y,z]; 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)+a*ln(beta*x)^n*diff(w(x,y,z),y)+ b*ln(lambda*y)^k*diff(w(x,y,z),z)= c*ln(gamma*x)^m*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!a \left ( \ln \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int ^{x}\!b \left ( \ln \left ( \lambda \, \left ( a\int \! \left ( \ln \left ( \beta \,{\it \_b} \right ) \right ) ^{n}\,{\rm d}{\it \_b}-\int \!a \left ( \ln \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{k}{d{\it \_b}}+z \right ) {{\rm e}^{\int \!c \left ( \ln \left ( x\gamma \right ) \right ) ^{m}\,{\rm d}x}}\]
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Added Oct 17, 2019.
Problem Chapter 8.5.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a_1 \ln ^{n_1}(\lambda _1 x) w_x + b_1 \ln ^{m_1}(\beta _1 y) w_y + c_1 \ln ^{k_1}(\gamma _1 z) w_z = \left ( a_2 \ln ^{n_2}(\lambda _2 x) + b_2 \ln ^{m_2}(\beta _2 y)+ c_2 \ln ^{k_2}(\gamma _2 z) \right ) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a1*Log[lambda1*x]^n1*D[w[x, y,z], x] + b1*Log[beta1*y]^m1*D[w[x, y,z], x]*D[w[x, y,z], y] + c1*Log[gamma1*z]^k1*D[w[x,y,z],z]== ( a2*Log[lambda2*x]^n2+ b2*Log[beta2*y]^m2+ c2*Log[gamma2*z]^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*ln(lambda1*x)^n1*diff(w(x,y,z),x)+ b1*ln(beta1*y)^m1*diff(w(x,y,z),y)+ c1*ln(gamma1*z)^k1*diff(w(x,y,z),z)= (a2*ln(lambda2*x)^n2+ b2*ln(beta2*y)^m2+ c2*ln(gamma2*z)^k2)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \! \left ( \ln \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( \ln \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}{\it a1}}{{\it b1}}}\,{\rm d}y,-\int \! \left ( \ln \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( \ln \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}{\it a1}}{{\it c1}}}\,{\rm d}z \right ) {{\rm e}^{\int ^{x}\!{\frac { \left ( \ln \left ( \lambda 1\,{\it \_f} \right ) \right ) ^{-{\it n1}}}{{\it a1}} \left ( {\it a2}\, \left ( \ln \left ( \lambda 2\,{\it \_f} \right ) \right ) ^{{\it n2}}+{\it b2}\, \left ( \ln \left ( \beta 2\,\RootOf \left ( \int \! \left ( \ln \left ( \lambda 1\,{\it \_f} \right ) \right ) ^{-{\it n1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \ln \left ( \beta 1\,{\it \_a} \right ) \right ) ^{-{\it m1}}{\it a1}}{{\it b1}}}{d{\it \_a}}-\int \! \left ( \ln \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( \ln \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}{\it a1}}{{\it b1}}}\,{\rm d}y \right ) \right ) \right ) ^{{\it m2}}+{\it c2}\, \left ( \ln \left ( \gamma 2\,\RootOf \left ( \int \! \left ( \ln \left ( \lambda 1\,{\it \_f} \right ) \right ) ^{-{\it n1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \ln \left ( \gamma 1\,{\it \_a} \right ) \right ) ^{-{\it k1}}{\it a1}}{{\it c1}}}{d{\it \_a}}-\int \! \left ( \ln \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( \ln \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}{\it a1}}{{\it c1}}}\,{\rm d}z \right ) \right ) \right ) ^{{\it k2}} \right ) }{d{\it \_f}}}}\]
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