6.3.13 5.1

6.3.13.1 [924] Problem 1

problem number 924

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'));
 

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

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6.3.13.2 [925] Problem 2

problem number 925

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'));
 

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

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6.3.13.3 [926] Problem 3

problem number 926

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]];
 

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

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 {-\left (\LambertW \left (\left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1} \lambda x \right )-\ln \left (\frac {\left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1} \lambda x}{\LambertW \left (\left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1} \lambda x \right )}\right )\right ) \left (\ln \left (\frac {\left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1} x}{\LambertW \left (\left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1} \lambda x \right )}\right )-\ln \left (\frac {\left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1} \lambda x}{\LambertW \left (\left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1} \lambda x \right )}\right )+\ln \left (\gamma \right )\right ) c \Ei \left (1, -\ln \left (\frac {\left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1} \lambda x}{\LambertW \left (\left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1} \lambda x \right )}\right )-1\right ) \LambertW \left (\left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1} \lambda x \right ) {\mathrm e}^{\LambertW \left (\left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1} \lambda x \right )-\ln \left (\frac {\left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1} \lambda x}{\LambertW \left (\left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1} \lambda x \right )}\right )}+\left (a \LambertW \left (\left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1} \lambda x \right ) \mathit {\_F1} \left (\frac {\left (\ln \left (\lambda x \right )-1\right ) b \beta x +a \Ei \left (1, -\ln \left (\beta y \right )\right )}{a \beta }\right )+\left (\ln \left (\lambda x \right )-1\right ) \left (\LambertW \left (\left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1} \lambda x \right )+\ln \left (\frac {\left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1} x}{\LambertW \left (\left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1} \lambda x \right )}\right )-\ln \left (\frac {\left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1} \lambda x}{\LambertW \left (\left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1} \lambda x \right )}\right )+\ln \left (\gamma \right )\right ) c x \right ) \lambda }{a \lambda \LambertW \left (\left (\ln \left (\lambda x \right )-1\right ) {\mathrm e}^{-1} \lambda x \right )}\]

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6.3.13.4 [927] Problem 4

problem number 927

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]];
 

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

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'));
 

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

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6.3.13.5 [928] Problem 5

problem number 928

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]];
 

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

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'));
 

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

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6.3.13.6 [929] Problem 6

problem number 929

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'));
 

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

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