6.3.13 5.1

6.3.13.1 [925] Problem 1

problem number 925

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["Global`*"]; 
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]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right )+\frac {c (a \beta y-b \beta x) \log (a (\beta y+\lambda x))}{a (a \lambda +b \beta )}+\frac {c x \log (\beta y+\lambda x)}{a}-\frac {c x}{a}\right \}\right \}\]

Maple ✓

restart; 
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'));
 

\[w \left ( x,y \right ) ={\frac {1}{a\lambda +b\beta } \left ( \left ( a\lambda +b\beta \right ) {\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) +c \left ( \ln \left ( \beta \,y+\lambda \,x \right ) -1 \right ) \left ( \beta \,y+\lambda \,x \right ) \right ) }\]

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6.3.13.2 [926] Problem 2

problem number 926

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["Global`*"]; 
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]];
 

\[\left \{\left \{w(x,y)\to \frac {a b c_1\left (y-\frac {b x}{a}\right )+a k y \log (\beta y)+b c x \log (\lambda x)-b c x-b k x}{a b}\right \}\right \}\]

Maple ✓

restart; 
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'));
 

\[w \left ( x,y \right ) ={\frac {1}{ab} \left ( cx\ln \left ( \lambda \,x \right ) b+aky\ln \left ( \beta \,y \right ) +{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) ba-aky-cxb \right ) }\]

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6.3.13.3 [927] Problem 3

problem number 927

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["Global`*"]; 
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]];
 

\[\left \{\left \{w(x,y)\to \frac {a b \beta \lambda c_1\left (\frac {\text {li}(\beta y)}{\beta }-\frac {b x (\log (\lambda x)-1)}{a}\right )+c \left (a \lambda \text {li}(\beta y)-b \beta \left (\log \left (e^{\text {ProductLog}\left (\frac {\lambda x (\log (\lambda x)-1)}{e}\right )+1}\right )-1\right ) \text {Ei}\left (\log \left (e^{\text {ProductLog}\left (\frac {\lambda x (\log (\lambda x)-1)}{e}\right )+1}\right )\right ) \left (\log \left (\frac {\gamma x (\log (\lambda x)-1)}{\text {ProductLog}\left (\frac {\lambda x (\log (\lambda x)-1)}{e}\right )}\right )-\log \left (\frac {\lambda x (\log (\lambda x)-1)}{\text {ProductLog}\left (\frac {\lambda x (\log (\lambda x)-1)}{e}\right )}\right )\right )+b \beta e^{\text {ProductLog}\left (\frac {\lambda x (\log (\lambda x)-1)}{e}\right )+1} \left (\log \left (\frac {\gamma x (\log (\lambda x)-1)}{\text {ProductLog}\left (\frac {\lambda x (\log (\lambda x)-1)}{e}\right )}\right )-\log \left (\frac {\lambda x (\log (\lambda x)-1)}{\text {ProductLog}\left (\frac {\lambda x (\log (\lambda x)-1)}{e}\right )}\right )\right )\right )+b \beta c \text {ProductLog}\left (\frac {\lambda x (\log (\lambda x)-1)}{e}\right ) \text {Ei}\left (\log \left (e^{\text {ProductLog}\left (\frac {\lambda x (\log (\lambda x)-1)}{e}\right )+1}\right )\right ) \left (\log \left (\frac {\gamma x (\log (\lambda x)-1)}{\text {ProductLog}\left (\frac {\lambda x (\log (\lambda x)-1)}{e}\right )}\right )-\log \left (\frac {\lambda x (\log (\lambda x)-1)}{\text {ProductLog}\left (\frac {\lambda x (\log (\lambda x)-1)}{e}\right )}\right )\right )}{a b \beta \lambda }\right \}\right \}\]

Maple ✓

restart; 
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'));
 

\[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 ) \Ei \left ( 1,-1-\ln \left ( {\frac {\lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) {{\rm e}^{-1}}}{\LambertW \left ( \lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) {{\rm e}^{-1}} \right ) }} \right ) \right ) c \left ( \ln \left ( {\frac {x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) {{\rm e}^{-1}}}{\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 ) {{\rm e}^{-1}}}{\LambertW \left ( \lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) {{\rm e}^{-1}} \right ) }} \right ) -3\,\ln \left ( 2 \right ) \right ) \left ( -\ln \left ( {\frac {\lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) {{\rm e}^{-1}}}{\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 ) {{\rm e}^{-\ln \left ( {\frac {\lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) {{\rm e}^{-1}}}{\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 ) }}+\lambda \, \left ( a{\it \_F1} \left ( {\frac {a\Ei \left ( 1,-\ln \left ( \beta \,y \right ) \right ) +\beta \,bx \left ( \ln \left ( \lambda \,x \right ) -1 \right ) }{a\beta }} \right ) \LambertW \left ( \lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) {{\rm e}^{-1}} \right ) +cx \left ( \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 ) {{\rm e}^{-1}}}{\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 ) {{\rm e}^{-1}}}{\LambertW \left ( \lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) {{\rm e}^{-1}} \right ) }} \right ) -3\,\ln \left ( 2 \right ) \right ) \left ( \ln \left ( \lambda \,x \right ) -1 \right ) \right ) \right ) }\]

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6.3.13.4 [928] Problem 4

problem number 928

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["Global`*"]; 
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]];
 

\[\left \{\left \{w(x,y)\to \int _1^x\frac {s \log ^k\left (\frac {\beta \left (-b \text {Gamma}(n+1,-\log (\lambda x)) \log ^n(\lambda x) (-\log (\lambda x))^{-n}+b \text {Gamma}(n+1,-\log (\lambda K[1])) (-\log (\lambda K[1]))^{-n} \log ^n(\lambda K[1])+a \lambda y\right )}{a \lambda }\right )+c \log ^m(\mu K[1])}{a}dK[1]+c_1\left (y-\frac {b (-\log (\lambda x))^{-n} \log ^n(\lambda x) \text {Gamma}(n+1,-\log (\lambda x))}{a \lambda }\right )\right \}\right \}\]

Maple ✓

restart; 
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'));
 

\[w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{a} \left ( c \left ( \ln \left ( \mu \,{\it \_b} \right ) \right ) ^{m}+s \left ( \ln \left ( {\frac {\beta }{a} \left ( b\int \! \left ( \ln \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b}+ \left ( -\int \!{\frac {b \left ( \ln \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) a \right ) } \right ) \right ) ^{k} \right ) }{d{\it \_b}}+{\it \_F1} \left ( -\int \!{\frac {b \left ( \ln \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) \]

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6.3.13.5 [929] Problem 5

problem number 929

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["Global`*"]; 
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]];
 

\[\left \{\left \{w(x,y)\to \int _1^y\frac {\log ^{-n}(\lambda K[1]) \left (s \log ^k(\beta K[1])+c \log ^m\left (\frac {\mu \left (-a \text {Gamma}(1-n,-\log (\lambda y)) (-\log (\lambda y))^n \log ^{-n}(\lambda y)+a \text {Gamma}(1-n,-\log (\lambda K[1])) (-\log (\lambda K[1]))^n \log ^{-n}(\lambda K[1])+b \lambda x\right )}{b \lambda }\right )\right )}{b}dK[1]+c_1\left (\frac {(-\log (\lambda y))^n \log ^{-n}(\lambda y) \text {Gamma}(1-n,-\log (\lambda y))}{\lambda }-\frac {b x}{a}\right )\right \}\right \}\]

Maple ✓

restart; 
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'));
 

\[w \left ( x,y \right ) =\int ^{y}\!{\frac { \left ( \ln \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{b} \left ( c \left ( \ln \left ( -{\frac {\mu \, \left ( a\int \! \left ( \ln \left ( y\lambda \right ) \right ) ^{-n}\,{\rm d}y-a\int \! \left ( \ln \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}\,{\rm d}{\it \_b}-bx \right ) }{b}} \right ) \right ) ^{m}+s \left ( \ln \left ( \beta \,{\it \_b} \right ) \right ) ^{k} \right ) }{d{\it \_b}}+{\it \_F1} \left ( -{\frac {a\int \! \left ( \ln \left ( y\lambda \right ) \right ) ^{-n}\,{\rm d}y}{b}}+x \right ) \]

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6.3.13.6 [930] Problem 6

problem number 930

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["Global`*"]; 
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]];
 

Failed

Maple ✓

restart; 
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'));
 

\[w \left ( x,y \right ) =\int \!{\frac {c \left ( -3\,\ln \left ( 2 \right ) +\ln \left ( x \right ) \right ) ^{m} \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-n}}{a}}\,{\rm d}x+{\it \_F1} \left ( -\int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-n}\,{\rm d}x+\int \!{\frac { \left ( \ln \left ( y\lambda \right ) \right ) ^{-k}a}{b}}\,{\rm d}y \right ) \]

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