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'));
\[w \left ( x,y,z \right ) =\int ^{x}\!{\frac {s \left ( \ln \left ( \beta \,{\it \_a} \right ) \right ) ^{m}}{a} \left ( {\frac {ya-b \left ( x-{\it \_a} \right ) }{a}} \right ) ^{m}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ya-bx}{a}},-\int ^{x}\!{\frac {c{{\it \_a}}^{n}}{a} \left ( \ln \left ( {\frac { \left ( ya-b \left ( x-{\it \_a} \right ) \right ) \lambda }{a}} \right ) \right ) ^{k}}{d{\it \_a}}+z \right ) \] 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]];
\[\left \{\left \{w(x,y,z)\to \frac {\left (m^2+3 m+2\right ) (n+1) c_1\left (\frac {-a x^{n+1}+n y+y}{n+1},\frac {-b x^{m+1}+m z+z}{m+1}\right )+\frac {(m+2) (-\log (\beta x))^{-r} (-\log (\lambda x))^{-k} \left (-\beta c (m+1) \left (a x^{n+1}-(n+1) y\right ) (-\log (\beta x))^r \log ^k(\lambda x) \text {Gamma}(k+1,-\log (\lambda x))-\lambda (n+1) s \left (b x^{m+1}-(m+1) z\right ) \log ^r(\beta x) (-\log (\lambda x))^k \text {Gamma}(r+1,-\log (\beta x))\right )}{\beta \lambda }+\frac {a c \left (m^2+3 m+2\right ) x^n (\lambda x)^{-n} \left (-\log ^2(\lambda x)\right )^k (-\log (\lambda x))^{-k} (-(n+2) \log (\lambda x))^{-k} \text {Gamma}(k+1,-(n+2) \log (\lambda x))}{\lambda ^2 (n+2)}+\frac {b (n+1) s x^m (\beta x)^{-m} (-\log (\beta x))^{-r} \left (-\log ^2(\beta x)\right )^r (-(m+2) \log (\beta x))^{-r} \text {Gamma}(r+1,-(m+2) \log (\beta x))}{\beta ^2}}{(m+1) (m+2) (n+1)}\right \}\right \}\]
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'));
\[w \left ( x,y,z \right ) =\int ^{x}\!{\frac { \left ( m+1 \right ) c \left ( a{{\it \_a}}^{n+1}-a{x}^{n+1}+yn+y \right ) \left ( \ln \left ( {\it \_a}\,\lambda \right ) \right ) ^{k}+s \left ( n+1 \right ) \left ( \ln \left ( \beta \,{\it \_a} \right ) \right ) ^{r} \left ( b{{\it \_a}}^{m+1}-b{x}^{m+1}+zm+z \right ) }{ \left ( m+1 \right ) \left ( n+1 \right ) }}{d{\it \_a}}+{\it \_F1} \left ( {\frac {-a{x}^{n+1}+y \left ( n+1 \right ) }{n+1}},{\frac {-b{x}^{m+1}+z \left ( m+1 \right ) }{m+1}} \right ) \] 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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\int _1^xa \log ^n(\lambda K[1])dK[1],z-\int _1^xb \left (y-\int _1^xa \log ^n(\lambda K[1])dK[1]+\int _1^{K[2]}a \log ^n(\lambda K[1])dK[1]\right ){}^mdK[2]\right )+\int _1^x\left (c \log ^k(\beta K[3])+s \log ^r\left (\gamma \left (z-\int _1^xb \left (y-\int _1^xa \log ^n(\lambda K[1])dK[1]+\int _1^{K[2]}a \log ^n(\lambda K[1])dK[1]\right ){}^mdK[2]+\int _1^{K[3]}b \left (y-\int _1^xa \log ^n(\lambda K[1])dK[1]+\int _1^{K[2]}a \log ^n(\lambda K[1])dK[1]\right ){}^mdK[2]\right )\right )\right )dK[3]\right \}\right \}\] 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'));
\[w \left ( x,y,z \right ) =\int ^{x}\!c \left ( \ln \left ( \beta \,{\it \_b} \right ) \right ) ^{k}+s \left ( -3\,\ln \left ( 2 \right ) +\ln \left ( {\frac {b \left ( a{\it \_b}- \left ( \ln \left ( {\it \_a}\,\lambda \right ) \right ) ^{-n}\int \!a \left ( \ln \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \left ( \ln \left ( {\it \_a}\,\lambda \right ) \right ) ^{-n} \right ) \left ( a \left ( \ln \left ( {\it \_a}\,\lambda \right ) \right ) ^{n}{\it \_b}-\int \!a \left ( \ln \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) ^{m}-b \left ( ax- \left ( \ln \left ( {\it \_a}\,\lambda \right ) \right ) ^{-n}\int \!a \left ( \ln \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \left ( \ln \left ( {\it \_a}\,\lambda \right ) \right ) ^{-n} \right ) \left ( a \left ( \ln \left ( {\it \_a}\,\lambda \right ) \right ) ^{n}x-\int \!a \left ( \ln \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) ^{m}+za \left ( m+1 \right ) }{a \left ( m+1 \right ) }} \right ) \right ) ^{r}{d{\it \_b}}+{\it \_F1} \left ( -\int \!a \left ( \ln \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y,{\frac {-b \left ( ax- \left ( \ln \left ( {\it \_a}\,\lambda \right ) \right ) ^{-n}\int \!a \left ( \ln \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \left ( \ln \left ( {\it \_a}\,\lambda \right ) \right ) ^{-n} \right ) \left ( a \left ( \ln \left ( {\it \_a}\,\lambda \right ) \right ) ^{n}x-\int \!a \left ( \ln \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) ^{m}+za \left ( m+1 \right ) }{a \left ( m+1 \right ) }} \right ) \] 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'));
\[w \left ( x,y,z \right ) =-\int ^{y}\!3\,{\frac {1}{\sqrt {2\,b\int \! \left ( \ln \left ( {\it \_f}\,\beta \right ) \right ) ^{k}\,{\rm d}{\it \_f}-2\,b\int \! \left ( \ln \left ( \beta \,y \right ) \right ) ^{k}\,{\rm d}y+{z}^{2}}} \left ( -1/3\,c \left ( \RootOf \left ( -\sqrt {2\,b \left ( \ln \left ( \beta \,{\it \_a} \right ) \right ) ^{k}y-2\,b\int \! \left ( \ln \left ( \beta \,y \right ) \right ) ^{k}\,{\rm d}y+{z}^{2}}-\int ^{{\it \_Z}}\!{\frac { \left ( \ln \left ( {\it \_a}\,\lambda \right ) \right ) ^{-n}}{a}}{d{\it \_a}} \left ( \ln \left ( \beta \,{\it \_a} \right ) \right ) ^{k}b+ \left ( \ln \left ( \beta \,{\it \_a} \right ) \right ) ^{k}\int \!{\frac { \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-n}}{a}}\,{\rm d}xb+\sqrt {2\,b \left ( \ln \left ( \beta \,{\it \_a} \right ) \right ) ^{k}{\it \_f}-2\,b\int \! \left ( \ln \left ( \beta \,y \right ) \right ) ^{k}\,{\rm d}y+{z}^{2}} \right ) \right ) ^{m}+s \left ( \ln \left ( 2 \right ) -1/3\,\ln \left ( {\it \_f} \right ) \right ) \right ) }{d{\it \_f}}+{\it \_F1} \left ( -2\,b\int \! \left ( \ln \left ( \beta \,y \right ) \right ) ^{k}\,{\rm d}y+{z}^{2},{\frac {1}{b} \left ( - \left ( \ln \left ( \beta \,{\it \_a} \right ) \right ) ^{-k}\sqrt {2\,b \left ( \ln \left ( \beta \,{\it \_a} \right ) \right ) ^{k}y-2\,b\int \! \left ( \ln \left ( \beta \,y \right ) \right ) ^{k}\,{\rm d}y+{z}^{2}}+\int \!{\frac { \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-n}}{a}}\,{\rm d}xb \right ) } \right ) \]
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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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b \log ^{1-n}(x)}{a (n-1)}-(m-1)^{\frac {1}{m-1}} \log (y) \left (\frac {(m-1)^{\frac {1}{1-m}}}{\log (y)}\right )^m,\frac {c \log ^{1-n}(x)}{a (n-1)}-(r-1)^{\frac {1}{r-1}} \log (z) \left (\frac {(r-1)^{\frac {1}{1-r}}}{\log (z)}\right )^r\right )-\frac {k (m-1)^{\frac {1}{m-1}} \log (y) \left (\frac {(m-1)^{\frac {1}{1-m}}}{\log (y)}\right )^m \log ^s\left (\exp \left (\left ((m-1)^{\frac {m}{m-1}} \log (y) \left (\frac {(m-1)^{\frac {1}{1-m}}}{\log (y)}\right )^m\right )^{\frac {1}{1-m}}\right )\right ) \left (\frac {\log \left (\exp \left (\left ((m-1)^{\frac {m}{m-1}} \log (y) \left (\frac {(m-1)^{\frac {1}{1-m}}}{\log (y)}\right )^m\right )^{\frac {1}{1-m}}\right )\right )}{\log \left (\exp \left (\left ((m-1)^{\frac {m}{m-1}} \log (y) \left (\frac {(m-1)^{\frac {1}{1-m}}}{\log (y)}\right )^m\right )^{\frac {1}{1-m}}\right )\right )-\left ((m-1)^{\frac {m}{m-1}} \log (y) \left (\frac {(m-1)^{\frac {1}{1-m}}}{\log (y)}\right )^m\right )^{\frac {1}{1-m}}}\right )^{-s} \, _2F_1\left (1-m,-s;2-m;-\frac {1}{\log \left (\exp \left (\left ((m-1)^{\frac {m}{m-1}} \left (\frac {(m-1)^{\frac {1}{1-m}}}{\log (y)}\right )^m \log (y)\right )^{\frac {1}{1-m}}\right )\right ) \left ((m-1)^{\frac {m}{m-1}} \left (\frac {(m-1)^{\frac {1}{1-m}}}{\log (y)}\right )^m \log (y)\right )^{\frac {1}{m-1}}-1}\right )}{b}\right \}\right \}\]
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'));
\[w \left ( x,y,z \right ) ={\frac {1}{a \left ( n-s-1 \right ) } \left ( a \left ( n-s-1 \right ) {\it \_F1} \left ( {\frac {b \left ( m-1 \right ) \left ( \ln \left ( x \right ) \right ) ^{-n+1}-a \left ( \ln \left ( y \right ) \right ) ^{-m+1} \left ( n-1 \right ) }{ \left ( n-1 \right ) b \left ( m-1 \right ) }},{\frac {c \left ( r-1 \right ) \left ( \ln \left ( x \right ) \right ) ^{-n+1}-a \left ( \ln \left ( z \right ) \right ) ^{-r+1} \left ( n-1 \right ) }{ \left ( n-1 \right ) c \left ( r-1 \right ) }} \right ) -k \left ( \ln \left ( x \right ) \right ) ^{s-n+1} \right ) }\]
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