Added April 5, 2019.
Problem Chapter 5.5.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b w_y = c w + \ln ^k(\lambda x) \ln ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+ Log[lambda*x]^k*Log[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} \log ^k(\lambda K[1]) \log ^n\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+ln(lambda*x)^k*ln(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac { \left ( \ln \left ( {\it \_a}\,\lambda \right ) \right ) ^{k}}{a} \left ( \ln \left ( {\frac { \left ( ya-b \left ( x-{\it \_a} \right ) \right ) \beta }{a}} \right ) \right ) ^{n}{{\rm e}^{-{\frac {{\it \_a}\,c}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \right ) {{\rm e}^{{\frac {cx}{a}}}}\]
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Added April 5, 2019.
Problem Chapter 5.5.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b w_y = c \ln ^k(\lambda x) w+ s \ln ^n(\beta x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*Log[lambda*x]^k*w[x,y]+s*Log[beta*x]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\frac {c (-\log (\lambda x))^{-k} \log ^k(\lambda x) \text {Gamma}(k+1,-\log (\lambda x))}{a \lambda }\right ) \left (\int _1^x\frac {\exp \left (-\frac {c \text {Gamma}(k+1,-\log (\lambda K[1])) (-\log (\lambda K[1]))^{-k} \log ^k(\lambda K[1])}{a \lambda }\right ) s \log ^n(\beta K[1])}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*ln(lambda*x)^k*w(x,y)+s*ln(beta*x)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int \!{\frac {s \left ( \ln \left ( \beta \,x \right ) \right ) ^{n}}{a}{{\rm e}^{-{\frac {c\int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x}{a}}}}}\,{\rm d}x+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \right ) {{\rm e}^{\int \!{\frac {c \left ( \ln \left ( \lambda \,x \right ) \right ) ^{k}}{a}}\,{\rm d}x}}\]
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Added April 5, 2019.
Problem Chapter 5.5.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b w_y = \left ( c_1 \ln ^{n_1}(\lambda _1 x) +c_2 \ln ^{n_2}(\lambda _2 y) \right ) w + s_1 \ln ^{k_1}(\beta _1 x)+s_2 \ln ^{k_2}(\beta _2 y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == ( c1*Log[lambda1*x]^n1 +c2*Log[lambda2*y]^n2)*w[x,y] + s1*Log[beta1*x]^k1+s2*Log[beta2*y]*k2; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\frac {\text {c1} \log ^{\text {n1}}(\text {lambda1} x) (-\log (\text {lambda1} x))^{-\text {n1}} \text {Gamma}(\text {n1}+1,-\log (\text {lambda1} x))}{a \text {lambda1}}+\frac {\text {c2} (-\log (\text {lambda2} y))^{-\text {n2}} \log ^{\text {n2}}(\text {lambda2} y) \text {Gamma}(\text {n2}+1,-\log (\text {lambda2} y))}{b \text {lambda2}}\right ) \left (\int _1^x\frac {\exp \left (-\frac {\text {c1} \text {Gamma}(\text {n1}+1,-\log (\text {lambda1} K[1])) \log ^{\text {n1}}(\text {lambda1} K[1]) (-\log (\text {lambda1} K[1]))^{-\text {n1}}}{a \text {lambda1}}-\frac {\text {c2} \text {Gamma}\left (\text {n2}+1,-\log \left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )\right ) \left (-\log \left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )\right )^{-\text {n2}} \log ^{\text {n2}}\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )}{b \text {lambda2}}\right ) \left (\text {s1} \log ^{\text {k1}}(\text {beta1} K[1])+\text {k2} \text {s2} \log \left (\text {beta2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = ( c1*ln(lambda1*x)^n1 +c2*ln(lambda2*y)^n2)*w(x,y) + s1*ln(beta1*x)^k1+s2*ln(beta2*y)*k2; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a}{{\rm e}^{-{\frac {1}{a}\int \!{\it c1}\, \left ( \ln \left ( \lambda 1\,{\it \_b} \right ) \right ) ^{{\it n1}}+{\it c2}\, \left ( \ln \left ( {\frac { \left ( ya-b \left ( x-{\it \_b} \right ) \right ) \lambda 2}{a}} \right ) \right ) ^{{\it n2}}\,{\rm d}{\it \_b}}}} \left ( {\it s2}\,\ln \left ( {\frac { \left ( ya-b \left ( x-{\it \_b} \right ) \right ) \beta 2}{a}} \right ) {\it k2}+{\it s1}\, \left ( \ln \left ( \beta 1\,{\it \_b} \right ) \right ) ^{{\it k1}} \right ) }{d{\it \_b}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{a} \left ( {\it c1}\, \left ( \ln \left ( \lambda 1\,{\it \_a} \right ) \right ) ^{{\it n1}}+{\it c2}\, \left ( \ln \left ( {\frac { \left ( ya-b \left ( x-{\it \_a} \right ) \right ) \lambda 2}{a}} \right ) \right ) ^{{\it n2}} \right ) }{d{\it \_a}}}}\]
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Added April 5, 2019.
Problem Chapter 5.5.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a \ln (\lambda x) w_x + b \ln (\mu y) w_y = c w + k \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*Log[lambda*x]*D[w[x, y], x] + b*Log[mu*y]*D[w[x, y], y] == c*w[x,y]+k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := a*ln(lambda*x)*diff(w(x,y),x)+ b*ln(mu*y)*diff(w(x,y),y) =c*w(x,y)+k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\frac {1}{c} \left ( {{\rm e}^{-{\frac {c\Ei \left ( 1,-\ln \left ( \lambda \,x \right ) \right ) }{a\lambda }}}}{\it \_F1} \left ( {\frac {\Ei \left ( 1,-\ln \left ( \lambda \,x \right ) \right ) b\mu -a\Ei \left ( 1,-\ln \left ( \mu \,y \right ) \right ) \lambda }{b\lambda \,\mu }} \right ) c-k \right ) }\]
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Added April 5, 2019.
Problem Chapter 5.5.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a \ln ^n(\lambda x) w_x + b \ln ^m(\mu x) w_y = c \ln ^k(\nu x) w + p \ln ^s(\beta y)+q \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Log[lambda*x]^n*D[w[x, y], x] + b*Log[mu*x]^m*D[w[x, y], y] == c*Log[nu*x]^k*w[x,y]+p*Log[beta*y]^s+q; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \log ^{-n}(\lambda K[2]) \log ^k(\nu K[2])}{a}dK[2]\right ) \left (c_1\left (y-\int _1^x\frac {b \log ^{-n}(\lambda K[1]) \log ^m(\mu K[1])}{a}dK[1]\right )+\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \log ^{-n}(\lambda K[2]) \log ^k(\nu K[2])}{a}dK[2]\right ) \log ^{-n}(\lambda K[3]) \left (p \log ^s\left (\beta \left (y-\int _1^x\frac {b \log ^{-n}(\lambda K[1]) \log ^m(\mu K[1])}{a}dK[1]+\int _1^{K[3]}\frac {b \log ^{-n}(\lambda K[1]) \log ^m(\mu K[1])}{a}dK[1]\right )\right )+q\right )}{a}dK[3]\right )\right \}\right \}\]
Maple ✓
restart; pde := a*ln(lambda*x)^n*diff(w(x,y),x)+ b*ln(mu*x)^m*diff(w(x,y),y) = c*ln(nu*x)^k*w(x,y)+p*ln(beta*y)^s+q; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac { \left ( \ln \left ( \lambda \,{\it \_f} \right ) \right ) ^{-n}}{a}{{\rm e}^{-{\frac {c\int \! \left ( \ln \left ( \nu \,{\it \_f} \right ) \right ) ^{k} \left ( \ln \left ( \lambda \,{\it \_f} \right ) \right ) ^{-n}\,{\rm d}{\it \_f}}{a}}}} \left ( p \left ( \ln \left ( {\frac {\beta \, \left ( ya-b\int \! \left ( \ln \left ( \mu \,x \right ) \right ) ^{m} \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-n}\,{\rm d}x+b\int \! \left ( \ln \left ( \mu \,{\it \_f} \right ) \right ) ^{m} \left ( \ln \left ( \lambda \,{\it \_f} \right ) \right ) ^{-n}\,{\rm d}{\it \_f} \right ) }{a}} \right ) \right ) ^{s}+q \right ) }{d{\it \_f}}+{\it \_F1} \left ( -{\frac {b\int \! \left ( \ln \left ( \mu \,x \right ) \right ) ^{m} \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-n}\,{\rm d}x}{a}}+y \right ) \right ) {{\rm e}^{\int \!{\frac { \left ( \ln \left ( \nu \,x \right ) \right ) ^{k}c \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-n}}{a}}\,{\rm d}x}}\]
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Added April 5, 2019.
Problem Chapter 5.5.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a \ln ^n(\lambda x) w_x + b \ln ^m(\mu x) w_y = c \ln ^k(\nu y) w + p \ln ^s(\beta x)+q \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Log[lambda*x]^n*D[w[x, y], x] + b*Log[mu*x]^m*D[w[x, y], y] == c*Log[nu*y]^k*w[x,y]+p*Log[beta*x]^s+q; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \log ^{-n}(\lambda K[2]) \log ^k\left (\nu \left (y-\int _1^x\frac {b \log ^{-n}(\lambda K[1]) \log ^m(\mu K[1])}{a}dK[1]+\int _1^{K[2]}\frac {b \log ^{-n}(\lambda K[1]) \log ^m(\mu K[1])}{a}dK[1]\right )\right )}{a}dK[2]\right ) \left (c_1\left (y-\int _1^x\frac {b \log ^{-n}(\lambda K[1]) \log ^m(\mu K[1])}{a}dK[1]\right )+\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \log ^{-n}(\lambda K[2]) \log ^k\left (\nu \left (y-\int _1^x\frac {b \log ^{-n}(\lambda K[1]) \log ^m(\mu K[1])}{a}dK[1]+\int _1^{K[2]}\frac {b \log ^{-n}(\lambda K[1]) \log ^m(\mu K[1])}{a}dK[1]\right )\right )}{a}dK[2]\right ) \left (p \log ^s(\beta K[3])+q\right ) \log ^{-n}(\lambda K[3])}{a}dK[3]\right )\right \}\right \}\]
Maple ✓
restart; pde := a*ln(lambda*x)^n*diff(w(x,y),x)+ b*ln(mu*x)^m*diff(w(x,y),y) = c*ln(nu*y)^k*w(x,y)+p*ln(beta*x)^s+q; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac { \left ( p \left ( \ln \left ( {\it \_f}\,\beta \right ) \right ) ^{s}+q \right ) \left ( \ln \left ( \lambda \,{\it \_f} \right ) \right ) ^{-n}}{a}{{\rm e}^{-{\frac {c}{a}\int \! \left ( \ln \left ( {\frac {\nu \, \left ( ya-b\int \! \left ( \ln \left ( \mu \,x \right ) \right ) ^{m} \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-n}\,{\rm d}x+b\int \! \left ( \ln \left ( \mu \,{\it \_f} \right ) \right ) ^{m} \left ( \ln \left ( \lambda \,{\it \_f} \right ) \right ) ^{-n}\,{\rm d}{\it \_f} \right ) }{a}} \right ) \right ) ^{k} \left ( \ln \left ( \lambda \,{\it \_f} \right ) \right ) ^{-n}\,{\rm d}{\it \_f}}}}}{d{\it \_f}}+{\it \_F1} \left ( -{\frac {b\int \! \left ( \ln \left ( \mu \,x \right ) \right ) ^{m} \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-n}\,{\rm d}x}{a}}+y \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {c \left ( \ln \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{a} \left ( \ln \left ( \nu \, \left ( \int \!{\frac {b \left ( \ln \left ( \mu \,{\it \_b} \right ) \right ) ^{m} \left ( \ln \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{a}}\,{\rm d}{\it \_b}-{\frac {b\int \! \left ( \ln \left ( \mu \,x \right ) \right ) ^{m} \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-n}\,{\rm d}x}{a}}+y \right ) \right ) \right ) ^{k}}{d{\it \_b}}}}\]
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