Added May 26, 2019.
Problem Chapter 6.5.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a x^n w_y + b \ln ^k(\lambda x) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*x^n*D[w[x, y,z], y] +b*Log[lambda*x]^k*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {-a x^{n+1}+n y+y}{n+1},z-\int _1^xb \log ^k(\lambda K[1])dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+ a*x^n*diff(w(x,y,z),y)+b*ln(lambda*x)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-xa{x}^{n}+y \left ( 1+n \right ) }{1+n}},-bx\ln \left ( x\lambda \right ) +xb+z \right ) \]
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Added May 26, 2019.
Problem Chapter 6.5.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a y + c \ln ^k(\lambda x)) w_y + (b z+ s \ln ^n(\lambda x)) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (a*y+c*Log[lambda*x]^k)*D[w[x, y,z], y] +(b*z+s*Log[lambda*x]^n)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y e^{-a x}-\int _1^xc e^{-a K[1]} (\log (\lambda )+\log (K[1]))^kdK[1],z e^{-b x}-\int _1^xe^{-b K[2]} s (\log (\lambda )+\log (K[2]))^ndK[2]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+ (a*y+c*ln(lambda*x)^k)*diff(w(x,y,z),y)+(b*z+s*log(lambda*x)^n)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -c\int \! \left ( \ln \left ( x\lambda \right ) \right ) ^{k}{{\rm e}^{-ax}}\,{\rm d}x+y{{\rm e}^{-ax}},-s\int \! \left ( \ln \left ( x\lambda \right ) \right ) ^{n}{{\rm e}^{-xb}}\,{\rm d}x+z{{\rm e}^{-xb}} \right ) \]
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Added May 26, 2019.
Problem Chapter 6.5.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a x w_x + b y w_y + (c \ln ^n(\lambda x)+ s \ln ^k(\beta y) ) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x*D[w[x, y,z], x] + b*y*D[w[x, y,z], y] +(c*Log[lambda*x]^n+s*Log[beta*y]^k)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y x^{-\frac {b}{a}},-\frac {c \log ^{n+1}(\lambda x)}{a n+a}-\frac {s \log ^{k+1}(\beta y)}{b k+b}+z\right )\right \}\right \}\]
Maple ✓
restart; pde := a*x*diff(w(x,y,z),x)+ b*y*diff(w(x,y,z),y)+(c*ln(lambda*x)^n+s*ln(beta*y)^k)*diff(w(x,y,z),z)= 0; 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 ( y{x}^{-{\frac {b}{a}}},{\frac {1}{ \left ( 2+2\,n \right ) ab \left ( 1+k \right ) } \left ( s \left ( -i\pi \, \left ( {\it csgn} \left ( iy \right ) -{\it csgn} \left ( i{x}^{{\frac {b}{a}}} \right ) \right ) {\it csgn} \left ( iy \right ) {\it csgn} \left ( iy{x}^{-{\frac {b}{a}}} \right ) -2\,\ln \left ( y{x}^{-{\frac {b}{a}}} \right ) -i\pi \, \left ( {\it csgn} \left ( iy \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{{\frac {b}{a}}} \right ) -2\,\ln \left ( {x}^{{\frac {b}{a}}} \right ) +i\pi \, \left ( {\it csgn} \left ( i\beta \,y \right ) \right ) ^{3}+ \left ( -i\pi \,{\it csgn} \left ( iy \right ) -i{\it csgn} \left ( i\beta \right ) \pi \right ) \left ( {\it csgn} \left ( i\beta \,y \right ) \right ) ^{2}+i\pi \,{\it csgn} \left ( iy \right ) {\it csgn} \left ( i\beta \right ) {\it csgn} \left ( i\beta \,y \right ) +i\pi \, \left ( {\it csgn} \left ( iy \right ) \right ) ^{3}-2\,\ln \left ( \beta \right ) \right ) a \left ( 1+n \right ) \left ( {\frac {i}{2}} \left ( {\it csgn} \left ( iy \right ) -{\it csgn} \left ( i{x}^{{\frac {b}{a}}} \right ) \right ) {\it csgn} \left ( iy \right ) \pi \,{\it csgn} \left ( iy{x}^{-{\frac {b}{a}}} \right ) +\ln \left ( y{x}^{-{\frac {b}{a}}} \right ) +{\frac {i}{2}} \left ( {\it csgn} \left ( iy \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{{\frac {b}{a}}} \right ) \pi +\ln \left ( {x}^{{\frac {b}{a}}} \right ) -{\frac {i}{2}} \left ( {\it csgn} \left ( i\beta \,y \right ) \right ) ^{3}\pi +{\frac { \left ( i{\it csgn} \left ( iy \right ) \pi +i{\it csgn} \left ( i\beta \right ) \pi \right ) \left ( {\it csgn} \left ( i\beta \,y \right ) \right ) ^{2}}{2}}-{\frac {i}{2}}{\it csgn} \left ( iy \right ) {\it csgn} \left ( i\beta \,y \right ) {\it csgn} \left ( i\beta \right ) \pi -{\frac {i}{2}} \left ( {\it csgn} \left ( iy \right ) \right ) ^{3}\pi +\ln \left ( \beta \right ) \right ) ^{k}+b \left ( 1+k \right ) \left ( \left ( i\pi \, \left ( {\it csgn} \left ( i\lambda \,x \right ) \right ) ^{3}-i\pi \, \left ( {\it csgn} \left ( ix \right ) +{\it csgn} \left ( i\lambda \right ) \right ) \left ( {\it csgn} \left ( i\lambda \,x \right ) \right ) ^{2}+i\pi \,{\it csgn} \left ( i\lambda \,x \right ) {\it csgn} \left ( i\lambda \right ) {\it csgn} \left ( ix \right ) -2\,\ln \left ( x \right ) -2\,\ln \left ( \lambda \right ) \right ) c \left ( -{\frac {i}{2}}\pi \, \left ( {\it csgn} \left ( i\lambda \,x \right ) \right ) ^{3}+{\frac {i}{2}}\pi \, \left ( {\it csgn} \left ( ix \right ) +{\it csgn} \left ( i\lambda \right ) \right ) \left ( {\it csgn} \left ( i\lambda \,x \right ) \right ) ^{2}-{\frac {i}{2}}\pi \,{\it csgn} \left ( i\lambda \,x \right ) {\it csgn} \left ( i\lambda \right ) {\it csgn} \left ( ix \right ) +\ln \left ( x \right ) +\ln \left ( \lambda \right ) \right ) ^{n}+2\,az \left ( 1+n \right ) \right ) \right ) } \right ) \]
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Added May 26, 2019.
Problem Chapter 6.5.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a x \ln (\lambda x) w_x + b y \ln (\beta y) w_y + c z \ln (\gamma z)w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x*Log[lambda*x]*D[w[x, y,z], x] + b*y*Log[beta*y]*D[w[x, y,z], y] +c*Log[gamma*z]*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\log \left ((\log (\beta )+\log (y)) (\log (\lambda )+\log (x))^{-\frac {b}{a}}\right ),\frac {\text {li}(\gamma z)}{\gamma }-\frac {c \log (\log (\lambda x))}{a}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*x*ln(lambda*x)*diff(w(x,y,z),x)+ b*y*ln(beta*y)*diff(w(x,y,z),y)+c*ln(gamma*z)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {\ln \left ( \ln \left ( x\lambda \right ) \right ) b-\ln \left ( \ln \left ( \beta \,y \right ) \right ) a}{a}},{\frac {-\ln \left ( \ln \left ( \beta \,y \right ) \right ) \gamma \,c-b\Ei \left ( 1,-\ln \left ( z \right ) -\ln \left ( \gamma \right ) \right ) }{c\gamma }} \right ) \]
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Added May 26, 2019.
Problem Chapter 6.5.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a x \ln (\lambda x) w_x + b y \ln (\beta y) w_y + c z \ln (\gamma x)w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x*Log[lambda*x]*D[w[x, y,z], x] + b*y*Log[beta*y]*D[w[x, y,z], y] +c*Log[gamma*x]*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {a z-c \log (\gamma x) \log (\log (\lambda x))+c \log (\lambda x) \log (\log (\lambda x))-c \log (x)}{a},\log \left ((\log (\beta )+\log (y)) (\log (\lambda )+\log (x))^{-\frac {b}{a}}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*x*ln(lambda*x)*diff(w(x,y,z),x)+ b*y*ln(beta*y)*diff(w(x,y,z),y)+c*ln(gamma*x)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {\ln \left ( \ln \left ( \beta \,y \right ) \right ) a-\ln \left ( \ln \left ( x\lambda \right ) \right ) b}{b}},{\frac {-c \left ( i\pi \, \left ( {\it csgn} \left ( i\lambda \,x \right ) \right ) ^{3}-i\pi \, \left ( {\it csgn} \left ( ix \right ) +{\it csgn} \left ( i\lambda \right ) \right ) \left ( {\it csgn} \left ( i\lambda \,x \right ) \right ) ^{2}+i\pi \,{\it csgn} \left ( i\lambda \,x \right ) {\it csgn} \left ( i\lambda \right ) {\it csgn} \left ( ix \right ) -2\,\ln \left ( \lambda \right ) +2\,\ln \left ( \gamma \right ) \right ) \ln \left ( -{\frac {i}{2}}\pi \, \left ( {\it csgn} \left ( i\lambda \,x \right ) \right ) ^{3}+{\frac {i}{2}}\pi \, \left ( {\it csgn} \left ( ix \right ) +{\it csgn} \left ( i\lambda \right ) \right ) \left ( {\it csgn} \left ( i\lambda \,x \right ) \right ) ^{2}-{\frac {i}{2}}\pi \,{\it csgn} \left ( i\lambda \right ) {\it csgn} \left ( ix \right ) {\it csgn} \left ( i\lambda \,x \right ) +\ln \left ( x \right ) +\ln \left ( \lambda \right ) \right ) +2\,za-2\,c\ln \left ( x \right ) }{2\,a}} \right ) \]
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Added May 26, 2019.
Problem Chapter 6.5.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a x \ln ^n(x) w_x + b y \ln ^m(y) w_y + c z \ln ^k(z)w_z = 0 \]
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]^k*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b \log ^{1-n}(x)}{a (n-1)}-(m-1)^{\frac {1}{m-1}} \log (y) \left (\frac {(m-1)^{\frac {1}{1-m}}}{\log (y)}\right )^m,\frac {c \log ^{1-n}(x)}{a (n-1)}-(k-1)^{\frac {1}{k-1}} \log (z) \left (\frac {(k-1)^{\frac {1}{1-k}}}{\log (z)}\right )^k\right )\right \}\right \}\]
Maple ✓
restart; 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)^k*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {b \left ( m-1 \right ) \left ( \ln \left ( x \right ) \right ) ^{1-n}-a \left ( \ln \left ( y \right ) \right ) ^{1-m} \left ( n-1 \right ) }{ \left ( n-1 \right ) b \left ( m-1 \right ) }},{\frac {c \left ( k-1 \right ) \left ( \ln \left ( x \right ) \right ) ^{1-n}-a \left ( \ln \left ( z \right ) \right ) ^{-k+1} \left ( n-1 \right ) }{ \left ( n-1 \right ) c \left ( k-1 \right ) }} \right ) \]
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