6.4.13 5.1

6.4.13.1 [1102] Problem 1

problem number 1102

Added Feb. 25, 2019.

Problem Chapter 4.5.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$a{w}_x+b{w}_y=c\ln (\lambda x+\beta y)w$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == c*Log[lambda*x + beta*y]*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde := a*diff(w(x,y),x)+b*diff(w(x,y),y) =  c*ln(lambda*x + beta*y)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

$$w(x,y)={(\beta y+\lambda x)}^{\frac{(\beta y+\lambda x)c}{a\lambda +b\beta }}\mathit{\_}\mathit{F}\mathit{1}\left(\frac{ay-bx}{a}\right){\mathrm{e}}^{-\frac{(\beta y+\lambda x)c}{a\lambda +b\beta }}$$

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6.4.13.2 [1103] Problem 2

problem number 1103

Added Feb. 25, 2019.

Problem Chapter 4.5.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$a{w}_x+b{w}_y=(c\ln (\lambda x)+k\ln (\beta y))w$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == (c*Log[lambda*x] + k*Log[beta*y])*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

$$\left\{\{w(x,y)\to e^{-\frac{x(c+k)}{a}}(\lambda x{)}^{\frac{cx}{a}}(\beta y{)}^{\frac{ky}{b}}{c}_1(y-\frac{bx}{a})\}\right\}$$

Maple ✓

restart; 
pde := a*diff(w(x,y),x)+b*diff(w(x,y),y) =  (c*ln(lambda*x)+k*ln(beta*y))*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

$$w(x,y)={\left(\beta y\right)}^{\frac{ky}{b}}{\left(\lambda x\right)}^{\frac{cx}{a}}\mathit{\_}\mathit{F}\mathit{1}\left(\frac{ay-bx}{a}\right){\mathrm{e}}^{\frac{-aky-bcx}{ab}}$$

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6.4.13.3 [1104] Problem 3

problem number 1104

Added Feb. 25, 2019.

Problem Chapter 4.5.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$a{w}_x+b{\ln }^n(\lambda x)w_y=(c{\ln }^m(\mu x)+s{\ln }^k(\beta y))w$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*Log[lambda*x]^n*D[w[x, y], y] == (c*Log[lambda*x]^m + s*Log[beta*y]^k)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde := a*diff(w(x,y),x)+b*ln(lambda*x)^n*diff(w(x,y),y) =  (c*ln(lambda*x)^m+s*ln(beta*y)^k)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

$$w(x,y)=\mathit{\_}\mathit{F}\mathit{1}(y-(\int \frac{b\ln {\left(\lambda x\right)}^n}{a}dx)){\mathrm{e}}^{{\int }_{}^x\frac{c\ln {\left(\mathit{\_}\mathit{b}\lambda \right)}^m+s\ln {\left((y+\int \frac{b\ln {\left(\mathit{\_}\mathit{b}\lambda \right)}^n}{a}d\mathit{\_}\mathit{b}-(\int \frac{b\ln {\left(\lambda x\right)}^n}{a}dx))\beta \right)}^k}{a}d\mathit{\_}\mathit{b}}$$

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6.4.13.4 [1105] Problem 4

problem number 1105

Added Feb. 25, 2019.

Problem Chapter 4.5.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$a{w}_x+b{\ln }^n(\lambda y)w_y=(c{\ln }^m(\mu x)+s{\ln }^k(\beta y))w$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*Log[lambda*y]^n*D[w[x, y], y] == (c*Log[lambda*x]^m + s*Log[beta*y]^k)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde := a*diff(w(x,y),x)+b*ln(lambda*y)^n*diff(w(x,y),y) =  (c*ln(lambda*x)^m+s*ln(beta*y)^k)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

$$w(x,y)=\mathit{\_}\mathit{F}\mathit{1}(-\frac{a(\int \ln {\left(\lambda y\right)}^{-n}dy)}{b}+x){\mathrm{e}}^{{\int }_{}^y\frac{(c\ln {\left((-\frac{a(\int \ln {\left(\lambda y\right)}^{-n}dy)}{b}+x+\int \frac{a\ln {\left(\mathit{\_}\mathit{b}\lambda \right)}^{-n}}{b}d\mathit{\_}\mathit{b})\lambda \right)}^m+s\ln {\left(\mathit{\_}\mathit{b}\beta \right)}^k)\ln {\left(\mathit{\_}\mathit{b}\lambda \right)}^{-n}}{b}d\mathit{\_}\mathit{b}}$$

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6.4.13.5 [1106] Problem 5

problem number 1106

Added Feb. 25, 2019.

Problem Chapter 4.5.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$\ln (\beta y)w_x+a\ln (\lambda x)w_y=bw\ln (\beta y)$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  Log[beta*y]*D[w[x, y], x] + a*Log[lambda*x]*D[w[x, y], y] == b*w[x, y]*Log[beta*y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

$$\left\{\{w(x,y)\to e^{bx}{c}_1\left(y(\log \left(\beta y{e}^{\frac{ax}{y}}{x}^{-\frac{ax}{y}}{\lambda }^{-\frac{ax}{y}}\right)-1)\right)\}\right\}$$

Maple ✓

restart; 
pde := ln(beta*y)*diff(w(x,y),x)+a*ln(lambda*x)*diff(w(x,y),y) =  b*w(x,y)*ln(beta*y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

$$w(x,y)=\mathit{\_}\mathit{F}\mathit{1}\left(\frac{-ax\ln \left(\lambda x\right)+ax+y\ln \left(\beta y\right)-y}{a}\right){\mathrm{e}}^{bx}$$

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6.4.13.6 [1107] Problem 6

problem number 1107

Added Feb. 25, 2019.

Problem Chapter 4.5.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$a\ln (\lambda x{)}^n{w}_x+b\ln (\beta y{)}^k{w}_y=c\ln (\gamma x{)}^{m}w$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a*Log[lambda*x]^n*D[w[x, y], x] + b*Log[beta*y]^k*D[w[x, y], y] == c*Log[gamma*x]^m*w[x, y]; 
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(beta*y)^k*diff(w(x,y),y) = c*log(gamma*x)^m*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

$$w(x,y)=\mathit{\_}\mathit{F}\mathit{1}(-(\int \ln {\left(\lambda x\right)}^{-n}dx)+\int \frac{a\ln {\left(\beta y\right)}^{-k}}{b}dy){\mathrm{e}}^{\int \frac{c{(\ln \left(x\right)+\ln \left(\gamma \right))}^m\ln {\left(\lambda x\right)}^{-n}}{a}dx}$$

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