Added June 26, 2019.
Problem Chapter 7.5.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c x^n \ln ^k(\lambda y) w_z = s y^m \ln ^r(\beta x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*x^n*Log[lambda*y]*D[w[x,y,z],z]== s*y^m*Log[beta*x]^r; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\frac {s \left (y+\frac {b (K[1]-x)}{a}\right )^m \log ^r(\beta K[1])}{a}dK[1]+c_1\left (y-\frac {b x}{a},\frac {b c x^{n+2} \, _2F_1\left (1,n+2;n+3;\frac {b x}{b x-a y}\right )}{a (n+1) (n+2) (a y-b x)}-\frac {c x^{n+1} \log (\lambda y)}{a n+a}+z\right )\right \}\right \}\]
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)^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int _{}^{x}\frac {s \left (\frac {a y -\left (-\textit {\_a} +x \right ) b}{a}\right )^{m} \ln \left (\textit {\_a} \beta \right )^{m}}{a}d \textit {\_a} +\textit {\_F1} \left (\frac {a y -b x}{a}, z -\left (\int _{}^{x}\frac {c \,\textit {\_a}^{n} \ln \left (\frac {\left (a y -\left (-\textit {\_a} +x \right ) b \right ) \lambda }{a}\right )^{k}}{a}d \textit {\_a} \right )\right )\] Answer has unresolved integrals
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Added June 26, 2019.
Problem Chapter 7.5.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a x^n w_y + b x^m w_z = c y \ln ^k(\lambda x)+s z \ln ^r(\beta x) \]
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; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \frac {\frac {(m+2) (-\log (\beta x))^{-r} (-\log (\lambda x))^{-k} \left (-\beta c (m+1) \left (a x^{n+1}-(n+1) y\right ) (-\log (\beta x))^r \log ^k(\lambda x) \operatorname {Gamma}(k+1,-\log (\lambda x))-\lambda (n+1) s \left (b x^{m+1}-(m+1) z\right ) \log ^r(\beta x) (-\log (\lambda x))^k \operatorname {Gamma}(r+1,-\log (\beta x))\right )}{\beta \lambda }+\left (m^2+3 m+2\right ) (n+1) c_1\left (\frac {-a x^{n+1}+n y+y}{n+1},\frac {-b x^{m+1}+m z+z}{m+1}\right )+\frac {a c \left (m^2+3 m+2\right ) x^n (\lambda x)^{-n} \left (-\log ^2(\lambda x)\right )^k (-\log (\lambda x))^{-k} (-((n+2) \log (\lambda x)))^{-k} \operatorname {Gamma}(k+1,-((n+2) \log (\lambda x)))}{\lambda ^2 (n+2)}+\frac {b (n+1) s x^m (\beta x)^{-m} (-\log (\beta x))^{-r} \left (-\log ^2(\beta x)\right )^r (-((m+2) \log (\beta x)))^{-r} \operatorname {Gamma}(r+1,-((m+2) \log (\beta x)))}{\beta ^2}}{(m+1) (m+2) (n+1)}\right \}\right \}\]
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; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int _{}^{x}\frac {\left (a \,\textit {\_a}^{n +1}-a \,x^{n +1}+n y +y \right ) \left (m +1\right ) c \ln \left (\textit {\_a} \lambda \right )^{k}+\left (n +1\right ) \left (b \,\textit {\_a}^{m +1}-b \,x^{m +1}+m z +z \right ) s \ln \left (\textit {\_a} \beta \right )^{r}}{\left (n +1\right ) \left (m +1\right )}d \textit {\_a} +\textit {\_F1} \left (\frac {-a \,x^{n +1}+\left (n +1\right ) y}{n +1}, \frac {-b \,x^{m +1}+\left (m +1\right ) z}{m +1}\right )\] Answer has unresolved integrals
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Added June 26, 2019.
Problem Chapter 7.5.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \ln ^n(\lambda x) w_y + b y^m w_z = c \ln ^k(\beta x)+s \ln ^r(\gamma z) \]
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; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\left (c \log ^k(\beta K[3])+s \log ^r\left (\gamma \left (z-\int _1^xb \left (y-\int _1^xa \log ^n(\lambda K[1])dK[1]+\int _1^{K[2]}a \log ^n(\lambda K[1])dK[1]\right ){}^mdK[2]+\int _1^{K[3]}b \left (y-\int _1^xa \log ^n(\lambda K[1])dK[1]+\int _1^{K[2]}a \log ^n(\lambda K[1])dK[1]\right ){}^mdK[2]\right )\right )\right )dK[3]+c_1\left (y-\int _1^xa \log ^n(\lambda K[1])dK[1],z-\int _1^xb \left (y-\int _1^xa \log ^n(\lambda K[1])dK[1]+\int _1^{K[2]}a \log ^n(\lambda K[1])dK[1]\right ){}^mdK[2]\right )\right \}\right \}\] Generated internal errors from solve : inconsistent or redundant transcendental equation
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; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int _{}^{x}\left (c \ln \left (\textit {\_b} \beta \right )^{k}+s \ln \left (-\frac {\left (-\left (m +1\right ) a z -\left (\textit {\_b} a +y \ln \left (\textit {\_a} \lambda \right )^{-n}-\ln \left (\textit {\_a} \lambda \right )^{-n} \left (\int a \ln \left (\lambda x \right )^{n}d x \right )\right ) b \left (\textit {\_b} a \ln \left (\textit {\_a} \lambda \right )^{n}+y -\left (\int a \ln \left (\lambda x \right )^{n}d x \right )\right )^{m}+\left (a x +y \ln \left (\textit {\_a} \lambda \right )^{-n}-\ln \left (\textit {\_a} \lambda \right )^{-n} \left (\int a \ln \left (\lambda x \right )^{n}d x \right )\right ) b \left (a x \ln \left (\textit {\_a} \lambda \right )^{n}+y -\left (\int a \ln \left (\lambda x \right )^{n}d x \right )\right )^{m}\right ) \gamma }{\left (m +1\right ) a}\right )^{r}\right )d \textit {\_b} +\textit {\_F1} \left (y -\left (\int a \ln \left (\lambda x \right )^{n}d x \right ), \frac {\left (m +1\right ) a z -\left (a x +y \ln \left (\textit {\_a} \lambda \right )^{-n}-\ln \left (\textit {\_a} \lambda \right )^{-n} \left (\int a \ln \left (\lambda x \right )^{n}d x \right )\right ) b \left (a x \ln \left (\textit {\_a} \lambda \right )^{n}+y -\left (\int a \ln \left (\lambda x \right )^{n}d x \right )\right )^{m}}{\left (m +1\right ) a}\right )\] Answer has unresolved integrals and RootOf
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Added June 26, 2019.
Problem Chapter 7.5.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a \ln ^n(\lambda x) w_x + z w_y + b \ln ^k(\beta y) w_z = c x^m +s \ln (\gamma y) \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*Log[lambda*x]^n*D[w[x, y,z], x] + z*D[w[x, y,z], y] + b*Log[beta*y]^k*D[w[x,y,z],z]== c*x^m+s*Log[gamma*y]; 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(beta*y)^k*diff(w(x,y,z),z)= c*x^m+s*ln(gamma*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int _{}^{y}\frac {c \RootOf \left (b \ln \left (\textit {\_a} \beta \right )^{k} \left (\int \frac {\ln \left (\lambda x \right )^{-n}}{a}d x \right )-b \ln \left (\textit {\_a} \beta \right )^{k} \left (\int _{}^{\textit {\_Z}}\frac {\ln \left (\textit {\_a} \lambda \right )^{-n}}{a}d \textit {\_a} \right )-\sqrt {2 b y \ln \left (\textit {\_a} \beta \right )^{k}-2 b \left (\int \ln \left (\beta y \right )^{k}d y \right )+z^{2}}+\sqrt {2 \textit {\_f} b \ln \left (\textit {\_a} \beta \right )^{k}-2 b \left (\int \ln \left (\beta y \right )^{k}d y \right )+z^{2}}\right )^{m}+s \ln \left (\textit {\_f} \gamma \right )}{\sqrt {2 b \left (\int \ln \left (\textit {\_f} \beta \right )^{k}d \textit {\_f} \right )-2 b \left (\int \ln \left (\beta y \right )^{k}d y \right )+z^{2}}}d \textit {\_f} +\textit {\_F1} \left (-2 b \left (\int \ln \left (\beta y \right )^{k}d y \right )+z^{2}, \frac {b \left (\int \frac {\ln \left (\lambda x \right )^{-n}}{a}d x \right )-\sqrt {2 b y \ln \left (\textit {\_a} \beta \right )^{k}-2 b \left (\int \ln \left (\beta y \right )^{k}d y \right )+z^{2}}\, \ln \left (\textit {\_a} \beta \right )^{-k}}{b}\right )\]
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Added June 26, 2019.
Problem Chapter 7.5.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a x \ln ^n(x) w_x + b y \ln ^m(y) w_y + c z \ln (z)^r w_z = k \ln ^s(x) \]
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[y]^s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to -\frac {k (m-1)^{\frac {1}{m-1}} \log (y) \left (\frac {(m-1)^{\frac {1}{1-m}}}{\log (y)}\right )^m \log ^s\left (\exp \left (\left ((m-1)^{\frac {m}{m-1}} \log (y) \left (\frac {(m-1)^{\frac {1}{1-m}}}{\log (y)}\right )^m\right )^{\frac {1}{1-m}}\right )\right ) \left (\frac {\log \left (\exp \left (\left ((m-1)^{\frac {m}{m-1}} \log (y) \left (\frac {(m-1)^{\frac {1}{1-m}}}{\log (y)}\right )^m\right )^{\frac {1}{1-m}}\right )\right )}{\log \left (\exp \left (\left ((m-1)^{\frac {m}{m-1}} \log (y) \left (\frac {(m-1)^{\frac {1}{1-m}}}{\log (y)}\right )^m\right )^{\frac {1}{1-m}}\right )\right )-\left ((m-1)^{\frac {m}{m-1}} \log (y) \left (\frac {(m-1)^{\frac {1}{1-m}}}{\log (y)}\right )^m\right )^{\frac {1}{1-m}}}\right )^{-s} \, _2F_1\left (1-m,-s;2-m;-\frac {1}{\log \left (\exp \left (\left ((m-1)^{\frac {m}{m-1}} \left (\frac {(m-1)^{\frac {1}{1-m}}}{\log (y)}\right )^m \log (y)\right )^{\frac {1}{1-m}}\right )\right ) \left ((m-1)^{\frac {m}{m-1}} \left (\frac {(m-1)^{\frac {1}{1-m}}}{\log (y)}\right )^m \log (y)\right )^{\frac {1}{m-1}}-1}\right )}{b}+c_1\left (\frac {b \log ^{1-n}(x)}{a (n-1)}-(m-1)^{\frac {1}{m-1}} \log (y) \left (\frac {(m-1)^{\frac {1}{1-m}}}{\log (y)}\right )^m,\frac {c \log ^{1-n}(x)}{a (n-1)}-(r-1)^{\frac {1}{r-1}} \log (z) \left (\frac {(r-1)^{\frac {1}{1-r}}}{\log (z)}\right )^r\right )\right \}\right \}\]
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; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {\left (n -s -1\right ) a \textit {\_F1} \left (\frac {-\left (n -1\right ) a \ln \left (y \right )^{-m +1}+\left (m -1\right ) b \ln \left (x \right )^{-n +1}}{\left (n -1\right ) \left (m -1\right ) b}, \frac {-\left (n -1\right ) a \ln \left (z \right )^{-r +1}+\left (r -1\right ) c \ln \left (x \right )^{-n +1}}{\left (r -1\right ) \left (n -1\right ) c}\right )-k \ln \left (x \right )^{-n +s +1}}{\left (n -s -1\right ) a}\]
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