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'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {a y -b x}{a}, z -\left (\int _{}^{x}\frac {c \mathit {\_a}^{n} \ln \left (\frac {\left (a y -\left (-\mathit {\_a} +x \right ) b \right ) \lambda }{a}\right )^{k}}{a}d\mathit {\_a} \right )\right ) {\mathrm e}^{\int _{}^{x}\frac {s \left (\frac {a y -\left (-\mathit {\_a} +x \right ) b}{a}\right )^{m} \ln \left (\mathit {\_a} \beta \right )^{r}}{a}d\mathit {\_a}}\]
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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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {-a x^{n+1}+n y+y}{n+1},\frac {-b x^{m+1}+m z+z}{m+1}\right ) \exp \left (\frac {\left (n^2+3 n+2\right ) (-\log (\beta x))^{-r} (-\log (\lambda x))^{-k} \left (\frac {\beta c \left (-a x^{n+1}+n y+y\right ) (-\log (\beta x))^r \log ^k(\lambda x) \operatorname {Gamma}(k+1,-\log (\lambda x))}{n+1}+\frac {\lambda s \left (-b x^{m+1}+m z+z\right ) \log ^r(\beta x) (-\log (\lambda x))^k \operatorname {Gamma}(r+1,-\log (\beta x))}{m+1}\right )}{\beta \lambda (n+1) (n+2)}+\frac {a c x^n (\lambda x)^{-n} \left (-\log ^2(\lambda x)\right )^k (-\log (\lambda x))^{-k} (-((n+2) \log (\lambda x)))^{-k} \operatorname {Gamma}(k+1,-((n+2) \log (\lambda x)))}{\lambda ^2 (n+1) (n+2)}+\frac {b s x^m (\beta x)^{-m} \log ^r(\beta x) (-((m+2) \log (\beta x)))^{-r} \operatorname {Gamma}(r+1,-((m+2) \log (\beta x)))}{\beta ^2 (m+1) (m+2)}\right )\right \}\right \}\]
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'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {-a x^{n +1}+\left (n +1\right ) y}{n +1}, \frac {-b x^{m +1}+\left (m +1\right ) z}{m +1}\right ) {\mathrm e}^{\int _{}^{x}\frac {\left (m +1\right ) \left (a \mathit {\_a}^{n +1}-a x^{n +1}+n y +y \right ) c \ln \left (\mathit {\_a} \lambda \right )^{k}+\left (n +1\right ) \left (b \mathit {\_a}^{m +1}-b x^{m +1}+m z +z \right ) s \ln \left (\mathit {\_a} \beta \right )^{r}}{\left (n +1\right ) \left (m +1\right )}d\mathit {\_a}}\]
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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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {a (-\log (\lambda x))^{-n} \log ^n(\lambda x) \operatorname {Gamma}(n+1,-\log (\lambda x))}{\lambda },z-\int _1^xb \left (\frac {-a \operatorname {Gamma}(n+1,-\log (\lambda x)) \log ^n(\lambda x) (-\log (\lambda x))^{-n}+a \operatorname {Gamma}(n+1,-\log (\lambda K[1])) (-\log (\lambda K[1]))^{-n} \log ^n(\lambda K[1])+\lambda y}{\lambda }\right )^mdK[1]\right ) \exp \left (\int _1^x\left (c \log ^k(\beta K[2])+s \log ^r\left (\gamma \left (z-\int _1^xb \left (\frac {-a \operatorname {Gamma}(n+1,-\log (\lambda x)) \log ^n(\lambda x) (-\log (\lambda x))^{-n}+a \operatorname {Gamma}(n+1,-\log (\lambda K[1])) (-\log (\lambda K[1]))^{-n} \log ^n(\lambda K[1])+\lambda y}{\lambda }\right )^mdK[1]+\int _1^{K[2]}b \left (\frac {-a \operatorname {Gamma}(n+1,-\log (\lambda x)) \log ^n(\lambda x) (-\log (\lambda x))^{-n}+a \operatorname {Gamma}(n+1,-\log (\lambda K[1])) (-\log (\lambda K[1]))^{-n} \log ^n(\lambda K[1])+\lambda y}{\lambda }\right )^mdK[1]\right )\right )\right )dK[2]\right )\right \}\right \}\]
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'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (-2 b \left (\int \ln \left (\lambda y \right )^{k}d y \right )+z^{2}, \frac {b \left (\int \frac {\ln \left (\lambda x \right )^{-n}}{a}d x \right )-\sqrt {2 b y \ln \left (\mathit {\_a} \lambda \right )^{k}-2 b \left (\int \ln \left (\lambda y \right )^{k}d y \right )+z^{2}}\, \ln \left (\mathit {\_a} \lambda \right )^{-k}}{b}\right ) {\mathrm e}^{\int _{}^{y}\frac {c \RootOf \left (b \ln \left (\mathit {\_a} \lambda \right )^{k} \left (\int \frac {\ln \left (\lambda x \right )^{-n}}{a}d x \right )-b \ln \left (\mathit {\_a} \lambda \right )^{k} \left (\int _{}^{\mathit {\_Z}}\frac {\ln \left (\mathit {\_a} \lambda \right )^{-n}}{a}d\mathit {\_a} \right )-\sqrt {2 b y \ln \left (\mathit {\_a} \lambda \right )^{k}-2 b \left (\int \ln \left (\lambda y \right )^{k}d y \right )+z^{2}}+\sqrt {2 \mathit {\_f} b \ln \left (\mathit {\_a} \lambda \right )^{k}-2 b \left (\int \ln \left (\lambda y \right )^{k}d y \right )+z^{2}}\right )^{m}+s \ln \left (\sqrt {2 b \left (\int \ln \left (\mathit {\_f} \lambda \right )^{k}d \mathit {\_f} \right )-2 b \left (\int \ln \left (\lambda y \right )^{k}d y \right )+z^{2}}\, \gamma \right )}{\sqrt {2 b \left (\int \ln \left (\mathit {\_f} \lambda \right )^{k}d \mathit {\_f} \right )-2 b \left (\int \ln \left (\lambda y \right )^{k}d y \right )+z^{2}}}d\mathit {\_f}}\]
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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'));
\[w \left (x , y , z\right ) = x^{-\frac {k \ln \left (x \right )^{-n +s}}{\left (n -s -1\right ) a}} \mathit {\_F1} \left (\frac {-\left (n -1\right ) a \ln \left (y \right )^{-m +1}+\left (m -1\right ) b \ln \left (x \right )^{-n +1}}{\left (n -1\right ) \left (m -1\right ) b}, \frac {-\left (n -1\right ) a \ln \left (z \right )^{-r +1}+\left (r -1\right ) c \ln \left (x \right )^{-n +1}}{\left (r -1\right ) \left (n -1\right ) c}\right )\]
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