Added Oct 17, 2019.
Problem Chapter 8.5.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b w_y + c x^n \ln ^k(\lambda y) w_z = s y^m \ln ^r(\beta x) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*x^n*Log[lambda*y]^k*D[w[x,y,z],z]== s*y^m*Log[beta*x]^r*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*diff(w(x,y,z),y)+ c*x^n*ln(lambda*y)^k*diff(w(x,y,z),z)= s*y^m*ln(beta*x)^r*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ay-xb}{a}},-\int ^{x}\!{\frac {c{{\it \_a}}^{n}}{a} \left ( \ln \left ( {\frac {\lambda \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{k}}{d{\it \_a}}+z \right ) {{\rm e}^{\int ^{x}\!{\frac {s \left ( \ln \left ( \beta \,{\it \_a} \right ) \right ) ^{r}}{a} \left ( {\frac {ay-b \left ( x-{\it \_a} \right ) }{a}} \right ) ^{m}}{d{\it \_a}}}}\]
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Added Oct 17, 2019.
Problem Chapter 8.5.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a x^n w_y + b x^m w_z = \left ( c y \ln ^k(\lambda x)+s z \ln ^r(\beta x)\right ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*x^n*D[w[x, y,z], y] + b*x^m*D[w[x,y,z],z]== (c*y*Log[lambda*x]^k+s*z*Log[beta*x]^r)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {-a x^{n+1}+n y+y}{n+1},\frac {-b x^{m+1}+m z+z}{m+1}\right ) \exp \left (\frac {c \log ^k(\lambda x) \left (\lambda \left (-a x^{n+1}+n y+y\right ) (-\log (\lambda x))^{-k} \text {Gamma}(k+1,-\log (\lambda x))+\frac {a x^n (\lambda x)^{-n} (-(n+2) \log (\lambda x))^{-k} \text {Gamma}(k+1,-(n+2) \log (\lambda x))}{n+2}\right )}{\lambda ^2 (n+1)}+\frac {b s x^m (\beta x)^{-m} \log ^r(\beta x) (-(m+2) \log (\beta x))^{-r} \text {Gamma}(r+1,-(m+2) \log (\beta x))}{\beta ^2 (m+1) (m+2)}+\frac {s \left (-b x^{m+1}+m z+z\right ) (-\log (\beta x))^{-r} \log ^r(\beta x) \text {Gamma}(r+1,-\log (\beta x))}{\beta (m+1)}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+a*x^n*diff(w(x,y,z),y)+ b*x^m*diff(w(x,y,z),z)= (c*y*ln(lambda*x)^k+s*z*ln(beta*x)^r)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-{x}^{1+n}a+y \left ( 1+n \right ) }{1+n}},{\frac {-{x}^{m+1}b+z \left ( m+1 \right ) }{m+1}} \right ) {{\rm e}^{\int ^{x}\!{\frac { \left ( {{\it \_a}}^{1+n}a-{x}^{1+n}a+yn+y \right ) c \left ( m+1 \right ) \left ( \ln \left ( {\it \_a}\,\lambda \right ) \right ) ^{k}+ \left ( {{\it \_a}}^{m+1}b-{x}^{m+1}b+zm+z \right ) s \left ( 1+n \right ) \left ( \ln \left ( \beta \,{\it \_a} \right ) \right ) ^{r}}{ \left ( 1+n \right ) \left ( m+1 \right ) }}{d{\it \_a}}}}\]
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Added Oct 17, 2019.
Problem Chapter 8.5.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a \ln ^n(\lambda x) w_y + b y^m w_z = \left ( c \ln ^k(\beta x)+s \ln ^r(\gamma z)\right ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Log[lambda*x]^n*D[w[x, y,z], y] + b*y^m*D[w[x,y,z],z]== (c*Log[beta*x]^k+s*Log[gamma*z]^r)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\int _1^xa \log ^n(\lambda K[1])dK[1],z-\int _1^xb \left (y-\int _1^xa \log ^n(\lambda K[1])dK[1]+\int _1^{K[2]}a \log ^n(\lambda K[1])dK[1]\right ){}^mdK[2]\right ) \exp \left (\int _1^x\left (c \log ^k(\beta K[3])+s \log ^r\left (\gamma \left (z-\int _1^xb \left (y-\int _1^xa \log ^n(\lambda K[1])dK[1]+\int _1^{K[2]}a \log ^n(\lambda K[1])dK[1]\right ){}^mdK[2]+\int _1^{K[3]}b \left (y-\int _1^xa \log ^n(\lambda K[1])dK[1]+\int _1^{K[2]}a \log ^n(\lambda K[1])dK[1]\right ){}^mdK[2]\right )\right )\right )dK[3]\right )\right \}\right \}\]
Maple ✗
restart; local gamma; pde := diff(w(x,y,z),x)+a*ln(lambda*x)^n*diff(w(x,y,z),y)+ b*y^m*diff(w(x,y,z),z)= (c*ln(beta*x)^k+s*ln(gamma*z)^r)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
server hangs
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Added Oct 17, 2019.
Problem Chapter 8.5.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a \ln ^n(\lambda x) w_x + z w_y + b \ln ^k(\beta y) w_z = \left ( c x^m +s \ln (\gamma y)\right ) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*Log[lambda*x]^n*D[w[x, y,z], x] + z*D[w[x, y,z], y] + b*Log[lambda*y]^k*D[w[x,y,z],z]== (c*x^m+s*Log[gamma*y])*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*ln(lambda*x)^n*diff(w(x,y,z),x)+z*diff(w(x,y,z),y)+ b*ln(lambda*y)^k*diff(w(x,y,z),z)= (c*x^m+s*ln(gamma*z))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -2\,b\int \! \left ( \ln \left ( \lambda \,y \right ) \right ) ^{k}\,{\rm d}y+{z}^{2},{\frac {1}{b} \left ( \int \!{\frac { \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-n}}{a}}\,{\rm d}xb- \left ( \ln \left ( \lambda \,{\it \_a} \right ) \right ) ^{-k}\sqrt {2\,b \left ( \ln \left ( \lambda \,{\it \_a} \right ) \right ) ^{k}y-2\,b\int \! \left ( \ln \left ( \lambda \,y \right ) \right ) ^{k}\,{\rm d}y+{z}^{2}} \right ) } \right ) {{\rm e}^{\int ^{y}\!{ \left ( c \left ( \RootOf \left ( -\sqrt {2\,b \left ( \ln \left ( \lambda \,{\it \_a} \right ) \right ) ^{k}y-2\,b\int \! \left ( \ln \left ( \lambda \,y \right ) \right ) ^{k}\,{\rm d}y+{z}^{2}}-\int ^{{\it \_Z}}\!{\frac { \left ( \ln \left ( \lambda \,{\it \_a} \right ) \right ) ^{-n}}{a}}{d{\it \_a}} \left ( \ln \left ( \lambda \,{\it \_a} \right ) \right ) ^{k}b+ \left ( \ln \left ( \lambda \,{\it \_a} \right ) \right ) ^{k}\int \!{\frac { \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-n}}{a}}\,{\rm d}xb+\sqrt {2\,b \left ( \ln \left ( \lambda \,{\it \_a} \right ) \right ) ^{k}{\it \_f}-2\,b\int \! \left ( \ln \left ( \lambda \,y \right ) \right ) ^{k}\,{\rm d}y+{z}^{2}} \right ) \right ) ^{m}+s\ln \left ( \gamma \,\sqrt {2\,b\int \! \left ( \ln \left ( \lambda \,{\it \_f} \right ) \right ) ^{k}\,{\rm d}{\it \_f}-2\,b\int \! \left ( \ln \left ( \lambda \,y \right ) \right ) ^{k}\,{\rm d}y+{z}^{2}} \right ) \right ) {\frac {1}{\sqrt {2\,b\int \! \left ( \ln \left ( \lambda \,{\it \_f} \right ) \right ) ^{k}\,{\rm d}{\it \_f}-2\,b\int \! \left ( \ln \left ( \lambda \,y \right ) \right ) ^{k}\,{\rm d}y+{z}^{2}}}}}{d{\it \_f}}}}\]
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Added Oct 17, 2019.
Problem Chapter 8.5.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a x (\ln x)^n w_x + b y (\ln y)^m w_y + c z(\ln z)^r w_z = k (\ln x)^s w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x*Log[x]^n*D[w[x, y,z], x] + b*y*Log[y]^m*D[w[x, y,z], y] + c*z*Log[z]^r*D[w[x,y,z],z]== k*Log[x]^s*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to e^{\frac {k \log ^{-n+s+1}(x)}{a (-n)+a s+a}} c_1\left (\frac {b \log ^{1-n}(x)}{a (n-1)}-(m-1)^{\frac {1}{m-1}} \log (y) \left (\frac {(m-1)^{\frac {1}{1-m}}}{\log (y)}\right )^m,\frac {c \log ^{1-n}(x)}{a (n-1)}-(r-1)^{\frac {1}{r-1}} \log (z) \left (\frac {(r-1)^{\frac {1}{1-r}}}{\log (z)}\right )^r\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*x*ln(x)^n*diff(w(x,y,z),x)+b*y*ln(y)^m*diff(w(x,y,z),y)+ c*z*ln(z)^r*diff(w(x,y,z),z)= k*ln(x)^s*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {b \left ( -1+m \right ) \left ( \ln \left ( x \right ) \right ) ^{-n+1}- \left ( \ln \left ( y \right ) \right ) ^{1-m}a \left ( n-1 \right ) }{ \left ( n-1 \right ) b \left ( -1+m \right ) }},{\frac {c \left ( -1+r \right ) \left ( \ln \left ( x \right ) \right ) ^{-n+1}- \left ( \ln \left ( z \right ) \right ) ^{1-r}a \left ( n-1 \right ) }{ \left ( n-1 \right ) c \left ( -1+r \right ) }} \right ) {x}^{-{\frac {k \left ( \ln \left ( x \right ) \right ) ^{s-n}}{a \left ( -1-s+n \right ) }}}\]
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