Added June 26, 2019.
Problem Chapter 7.5.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a w_y + b w_z = c \ln ^k(\lambda x)+s \]
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[lambda*x]^k+s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x)+\frac {c \log ^k(\lambda x) (-\log (\lambda x))^{-k} \text {Gamma}(k+1,-\log (\lambda x))}{\lambda }+s x\right \}\right \}\]
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*ln(lambda*x)^k+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int \!c \left ( \ln \left ( x\lambda \right ) \right ) ^{k}\,{\rm d}x+sx+{\it \_F1} \left ( -ax+y,-xb+z \right ) \] Contains unresolve integral because maple can not integrate \(\ln ^n(x)\)
____________________________________________________________________________________
Added June 26, 2019.
Problem Chapter 7.5.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \ln (\beta y) \ln (\gamma z) w_z = k \ln (\alpha x) \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*Log[beta*y]*Log[gamma*z]*D[w[x,y,z],z]== k*Log[alpha*x]; 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*diff(w(x,y,z),y)+ c*ln(beta*y)*ln(gamma*z)*diff(w(x,y,z),z)=k*ln(alpha*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\frac {1}{a} \left ( {\it \_F1} \left ( {\frac {-ay+xb}{b}},{\frac {-b\Ei \left ( 1,-\ln \left ( \gamma \,z \right ) \right ) -c\gamma \,y \left ( \ln \left ( \beta \,y \right ) -1 \right ) }{c\gamma }} \right ) a+xk \left ( \ln \left ( \alpha \,x \right ) -1 \right ) \right ) }\]
____________________________________________________________________________________
Added June 26, 2019.
Problem Chapter 7.5.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \ln ^n(\beta x) w_y + b \ln ^k(\lambda x) w_z = c \ln ^m(\gamma x)+s \]
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]+s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to x (c \log (\gamma x)-c+s)+c_1\left (y-\frac {a (-\log (\beta x))^{-n} \log ^n(\beta x) \text {Gamma}(n+1,-\log (\beta x))}{\beta },z-\int _1^xb \log ^k(\lambda K[1])dK[1]\right )\right \}\right \}\]
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)+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!a \left ( \ln \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( \ln \left ( x\lambda \right ) \right ) ^{k}\,{\rm d}x+z \right ) +cx\ln \left ( x\gamma \right ) + \left ( -c+s \right ) x\] Contains unresolve integral because maple can not integrate \(\ln ^n(x)\)
____________________________________________________________________________________
Added June 26, 2019.
Problem Chapter 7.5.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \ln ^n(\lambda x) w_y + b \ln ^m(\beta y) w_z = c \ln ^k(\gamma y)+s \ln ^r(\mu z) \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Log[lambda*x]^n*D[w[x, y,z], y] + b*Log[beta*y]^m*D[w[x,y,z],z]== c*Log[gamma*y]^k+s*Log[mu*z]^r; 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(lambda*x)^n*diff(w(x,y,z),y)+ b*ln(beta*y)^m*diff(w(x,y,z),z)=c*ln(gamma*y)^k+s*ln(mu*z)^r; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int ^{x}\!c \left ( \ln \left ( \gamma \, \left ( a\int \! \left ( \ln \left ( {\it \_f}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_f}-\int \!a \left ( \ln \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{k}+s \left ( \ln \left ( \mu \, \left ( b\int \! \left ( \ln \left ( \beta \, \left ( a\int \! \left ( \ln \left ( {\it \_f}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_f}-\int \!a \left ( \ln \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{m}\,{\rm d}{\it \_f}-\int ^{x}\!b \left ( \ln \left ( \beta \, \left ( a\int \! \left ( \ln \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b}-\int \!a \left ( \ln \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{m}{d{\it \_b}}+z \right ) \right ) \right ) ^{r}{d{\it \_f}}+{\it \_F1} \left ( -\int \!a \left ( \ln \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y,-\int ^{x}\!b \left ( \ln \left ( \beta \, \left ( a\int \! \left ( \ln \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b}-\int \!a \left ( \ln \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{m}{d{\it \_b}}+z \right ) \]
____________________________________________________________________________________
Added June 26, 2019.
Problem Chapter 7.5.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a_1 \ln ^{n_1}(\lambda _1 x) w_x + b_1 \ln ^{m_1}(\beta _1 y) w_y + c_1 \ln ^{k_1}(\gamma _1 z) w_z = a_2 \ln ^{n_2}(\lambda _2 x)+ b_2 \ln ^{m_2}(\beta _2 y) + c_2 \ln ^{k_2}(\gamma _2 z) \]
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], y] + c1*Log[gamma1*z]^k1*D[w[x,y,z],z]== a2*Log[lambda2*x]^n2*D[w[x, y,z], x] + b2*Log[beta2*y]^m2+ c2*Log[gamma2*z]^k2; 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; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) =\int ^{x}\!{\frac { \left ( \ln \left ( \lambda 1\,{\it \_f} \right ) \right ) ^{-{\it n1}}}{{\it a1}} \left ( {\it a2}\, \left ( \ln \left ( \lambda 2\,{\it \_f} \right ) \right ) ^{{\it n2}}+{\it b2}\, \left ( \ln \left ( \beta 2\,\RootOf \left ( \int \! \left ( \ln \left ( \lambda 1\,{\it \_f} \right ) \right ) ^{-{\it n1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \ln \left ( \beta 1\,{\it \_a} \right ) \right ) ^{-{\it m1}}{\it a1}}{{\it b1}}}{d{\it \_a}}-\int \! \left ( \ln \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( \ln \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}{\it a1}}{{\it b1}}}\,{\rm d}y \right ) \right ) \right ) ^{{\it m2}}+{\it c2}\, \left ( \ln \left ( \gamma 2\,\RootOf \left ( \int \! \left ( \ln \left ( \lambda 1\,{\it \_f} \right ) \right ) ^{-{\it n1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \ln \left ( \gamma 1\,{\it \_a} \right ) \right ) ^{-{\it k1}}{\it a1}}{{\it c1}}}{d{\it \_a}}-\int \! \left ( \ln \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( \ln \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}{\it a1}}{{\it c1}}}\,{\rm d}z \right ) \right ) \right ) ^{{\it k2}} \right ) }{d{\it \_f}}+{\it \_F1} \left ( -\int \! \left ( \ln \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( \ln \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}{\it a1}}{{\it b1}}}\,{\rm d}y,-\int \! \left ( \ln \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}\,{\rm d}x+\int \!{\frac { \left ( \ln \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}{\it a1}}{{\it c1}}}\,{\rm d}z \right ) \] Contains RootOf and unresolved integrals \(\ln ^n(x)\)
____________________________________________________________________________________