____________________________________________________________________________________
Added Feb. 11, 2019.
Problem Chapter 3.5.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c \ln (\lambda x+\beta y) \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*Log[lambda*x + beta*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to \frac {a^2 \lambda c_1\left (\frac {a y-b x}{a}\right )+a \beta c y \log (\beta (a y-b x)+a \lambda x+b \beta x)+a c \lambda x \log \left (\frac {\beta (a y-b x)}{a}+\frac {b \beta x}{a}+\lambda x\right )-b \beta c x \log (\beta (a y-b x)+a \lambda x+b \beta x)+b \beta c x \log \left (\frac {\beta (a y-b x)}{a}+\frac {b \beta x}{a}+\lambda x\right )+a b \beta c_1\left (\frac {a y-b x}{a}\right )-a c \lambda x-b \beta c x}{a (a \lambda +b \beta )}\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; pde :=a*diff(w(x,y),x) + b*diff(w(x,y),y) = c*ln(lambda*x+beta*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) ={\frac {x}{a\lambda +b\beta } \left ( \ln \left ( {\frac { \left ( a\lambda +b\beta \right ) x}{a}}+{\frac { \left ( ya-bx \right ) \beta }{a}} \right ) c\lambda -\lambda \,c \right ) }+{\frac {y}{a\lambda +b\beta } \left ( c\ln \left ( {\frac { \left ( a\lambda +b\beta \right ) x}{a}}+{\frac { \left ( ya-bx \right ) \beta }{a}} \right ) \beta -\beta \,c \right ) }+{\frac {1}{a\lambda +b\beta } \left ( {\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) a\lambda +{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) b\beta \right ) } \]
____________________________________________________________________________________
Added Feb. 11, 2019.
Problem Chapter 3.5.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c \ln (\lambda x) + k \ln (\beta y) \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*Log[lambda*x] + k*Log[beta*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to \frac {b k x \log \left (\beta \left (\frac {a y-b x}{a}+\frac {b x}{a}\right )\right )+a b c_1\left (\frac {a y-b x}{a}\right )-b k x \log (a y)+a k y \log (a y)+b c x \log (\lambda x)-b c x-b k x}{a b}\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; pde :=a*diff(w(x,y),x) + b*diff(w(x,y),y) = c*ln(lambda*x)+k*ln(beta*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) ={\frac { \left ( \ln \left ( \lambda \,x \right ) bc-bc \right ) x}{ab}}+{\frac {y}{ab} \left ( \ln \left ( {\frac {bx\beta }{a}}+{\frac { \left ( ya-bx \right ) \beta }{a}} \right ) ak-ak \right ) }+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \]
____________________________________________________________________________________
Added Feb. 11, 2019.
Problem Chapter 3.5.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \ln (\lambda x) \ln (\beta y) w_y = c \ln (\gamma x) \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; pde = a*D[w[x, y], x] + b*Log[lambda*x]*Log[beta*y]*D[w[x, y], y] == c*Log[gamma*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y)\to \frac {c \left (a \lambda \text {LogIntegral}(\beta y)-b \beta \left (\log \left (e^{\text {ProductLog}\left (\frac {\lambda x (\log (\lambda x)-1)}{e}\right )+1}\right )-1\right ) \text {ExpIntegralEi}\left (\log \left (e^{\text {ProductLog}\left (\frac {\lambda x (\log (\lambda x)-1)}{e}\right )+1}\right )\right ) \left (\log \left (\frac {\gamma x (\log (\lambda x)-1)}{\text {ProductLog}\left (\frac {\lambda x (\log (\lambda x)-1)}{e}\right )}\right )-\log \left (\frac {\lambda x (\log (\lambda x)-1)}{\text {ProductLog}\left (\frac {\lambda x (\log (\lambda x)-1)}{e}\right )}\right )\right )+b \beta e^{\text {ProductLog}\left (\frac {\lambda x (\log (\lambda x)-1)}{e}\right )+1} \left (\log \left (\frac {\gamma x (\log (\lambda x)-1)}{\text {ProductLog}\left (\frac {\lambda x (\log (\lambda x)-1)}{e}\right )}\right )-\log \left (\frac {\lambda x (\log (\lambda x)-1)}{\text {ProductLog}\left (\frac {\lambda x (\log (\lambda x)-1)}{e}\right )}\right )\right )\right )+a b \beta \lambda c_1\left (\frac {\text {LogIntegral}(\beta y)}{\beta }-\frac {b x (\log (\lambda x)-1)}{a}\right )+b \beta c \text {ProductLog}\left (\frac {\lambda x (\log (\lambda x)-1)}{e}\right ) \text {ExpIntegralEi}\left (\log \left (e^{\text {ProductLog}\left (\frac {\lambda x (\log (\lambda x)-1)}{e}\right )+1}\right )\right ) \left (\log \left (\frac {\gamma x (\log (\lambda x)-1)}{\text {ProductLog}\left (\frac {\lambda x (\log (\lambda x)-1)}{e}\right )}\right )-\log \left (\frac {\lambda x (\log (\lambda x)-1)}{\text {ProductLog}\left (\frac {\lambda x (\log (\lambda x)-1)}{e}\right )}\right )\right )}{a b \beta \lambda }\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; pde :=a*diff(w(x,y),x) + b*ln(lambda*x)*ln(beta*y)*diff(w(x,y),y) = c*ln(gamma*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime')); sol:=simplify(sol);
\[ w \left ( x,y \right ) ={\frac {1}{a\lambda \,\LambertW \left ( \lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) {{\rm e}^{-1}} \right ) } \left ( \LambertW \left ( \lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) {{\rm e}^{-1}} \right ) {\it \_F1} \left ( {\frac {\Ei \left ( 1,-\ln \left ( \beta \,y \right ) \right ) a+bx\beta \, \left ( \ln \left ( \lambda \,x \right ) -1 \right ) }{a\beta }} \right ) a\lambda + \left ( -\LambertW \left ( \lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) {{\rm e}^{-1}} \right ) \left ( 1-\ln \left ( {\frac {\lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) }{\LambertW \left ( \lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) {{\rm e}^{-1}} \right ) }} \right ) +\LambertW \left ( \lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) {{\rm e}^{-1}} \right ) \right ) \left ( \ln \left ( \gamma \right ) +\ln \left ( {\frac {x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) }{\LambertW \left ( \lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) {{\rm e}^{-1}} \right ) }} \right ) -\ln \left ( {\frac {\lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) }{\LambertW \left ( \lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) {{\rm e}^{-1}} \right ) }} \right ) \right ) \Ei \left ( 1,-\ln \left ( {\frac {\lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) }{\LambertW \left ( \lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) {{\rm e}^{-1}} \right ) }} \right ) \right ) +\lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) \left ( \ln \left ( \gamma \right ) +\LambertW \left ( \lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) {{\rm e}^{-1}} \right ) +\ln \left ( {\frac {x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) }{\LambertW \left ( \lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) {{\rm e}^{-1}} \right ) }} \right ) -\ln \left ( {\frac {\lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) }{\LambertW \left ( \lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) {{\rm e}^{-1}} \right ) }} \right ) \right ) \right ) c \right ) } \]
____________________________________________________________________________________
Added Feb. 11, 2019.
Problem Chapter 3.5.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \ln ^n(\lambda x) w_y = c \ln ^m(\mu x)+ s \ln ^k(\beta y) \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; pde = a*D[w[x, y], x] + b*Log[lambda*x]^n*D[w[x, y], y] == c*Log[mu*x]^m + s*Log[beta*y]^k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {\$Aborted} \] Timed out
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; pde :=a*diff(w(x,y),x) + b*ln(lambda*x)^n*diff(w(x,y),y) = c*ln(mu*x)^m+s*ln(beta*y)^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{a} \left ( c \left ( \ln \left ( {\it \_b}\,\mu \right ) \right ) ^{m}+s \left ( \ln \left ( {\frac {\beta }{a} \left ( b\int \! \left ( \ln \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b}+ \left ( -\int \!{\frac {b \left ( \ln \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) a \right ) } \right ) \right ) ^{k} \right ) }{d{\it \_b}}+{\it \_F1} \left ( -\int \!{\frac {b \left ( \ln \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) \]
____________________________________________________________________________________
Added Feb. 11, 2019.
Problem Chapter 3.5.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \ln ^n(\lambda y) w_y = c \ln ^m(\mu x)+ s \ln ^k(\beta y) \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; pde = a*D[w[x, y], x] + b*Log[lambda*y]^n*D[w[x, y], y] == c*Log[mu*x]^m + s*Log[beta*y]^k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {\$Aborted} \] Timed out
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; pde :=a*diff(w(x,y),x) + b*ln(lambda*y)^n*diff(w(x,y),y) = c*ln(mu*x)^m+s*ln(beta*y)^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) =\int ^{y}\!{\frac { \left ( \ln \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{b} \left ( c \left ( \ln \left ( {\frac {\mu }{b} \left ( a\int \! \left ( \ln \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}\,{\rm d}{\it \_b}+ \left ( -{\frac {a\int \! \left ( \ln \left ( y\lambda \right ) \right ) ^{-n}\,{\rm d}y}{b}}+x \right ) b \right ) } \right ) \right ) ^{m}+s \left ( \ln \left ( \beta \,{\it \_b} \right ) \right ) ^{k} \right ) }{d{\it \_b}}+{\it \_F1} \left ( -{\frac {a\int \! \left ( \ln \left ( y\lambda \right ) \right ) ^{-n}\,{\rm d}y}{b}}+x \right ) \]
____________________________________________________________________________________
Added Feb. 11, 2019.
Problem Chapter 3.5.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a \ln ^n(\lambda x) w_x + b \ln ^k(\beta y) w_y = c \ln ^m(\gamma x) \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; pde = a*Log[lambda*x]^n*D[w[x, y], x] + b*Log[lambda*y]^k*D[w[x, y], y] == c*Log[gamma*x]^m; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; pde :=a*ln(lambda*x)^n*diff(w(x,y),x) + b*ln(lambda*y)^k*diff(w(x,y),y) = c*ln(gamma*x)^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) =\int \!{\frac {c \left ( \ln \left ( \gamma \right ) +\ln \left ( x \right ) \right ) ^{m} \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-n}}{a}}\,{\rm d}x+{\it \_F1} \left ( -\int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-n}\,{\rm d}x+\int \!{\frac { \left ( \ln \left ( y\lambda \right ) \right ) ^{-k}a}{b}}\,{\rm d}y \right ) \]