7.4.14 5.2

7.4.14.1 [1110] Problem 1

problem number 1110

Added Feb. 25, 2019.

Problem Chapter 4.5.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

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

Mathematica ✓

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

$$\left\{\{w(x,y)\to c_1(y-\frac{bx}{a})\exp (\frac{s{\log }^k(\gamma y)(-\log (\gamma y){)}^{-k}\mathrm{Gamma}(k+1,-\log (\gamma y))}{b\gamma }+\frac{c{x}^{n+1}}{an+a})\}\right\}$$

Maple ✓

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

$$w(x,y)=\mathit{\text{\_F1}}\left(\frac{ay-bx}{a}\right){\mathrm{e}}^{{\int }_{}^x\frac{c{\mathit{\text{\_a}}}^n+s{(\ln \left(\frac{ay-(-\mathit{\text{\_a}}+x)b}{a}\right)+\ln \left(\gamma \right))}^k}{a}d\mathit{\text{\_a}}}$$

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7.4.14.2 [1111] Problem 2

problem number 1111

Added Feb. 25, 2019.

Problem Chapter 4.5.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$w_x+a{w}_y=(b{y}^2+c{x}^{n}y+s{\ln }^k(\lambda x))w$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + a*D[w[x, y], y] == (b*y^2 + c*x^n*y + s*Log[lambda*x]^k)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

$$\left\{\{w(x,y)\to c_1(y-ax)\exp (\frac{s{\log }^k(\lambda x)(-\log (\lambda x){)}^{-k}\mathrm{Gamma}(k+1,-\log (\lambda x))}{\lambda }+\frac{1}{3}{a}^{2}b{x}^3+ab{x}^2(y-ax)+bx(y-ax{)}^2+\frac{c{x}^{n+1}((n+2)y-ax)}{(n+1)(n+2)})\}\right\}$$

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+a*diff(w(x,y),y) = (b*y^2+c*x^n*y+ s*ln(lambda*x)^k)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

$$w(x,y)=\mathit{\text{\_F1}}(-ax+y){\mathrm{e}}^{{\int }_{}^x(ac{\mathit{\text{\_a}}}^{n+1}-(ax-y)c{\mathit{\text{\_a}}}^n+s\ln {\left(\mathit{\text{\_a}}\lambda \right)}^k+{((-\mathit{\text{\_a}}+x)a-y)}^{2}b)d\mathit{\text{\_a}}}$$

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7.4.14.3 [1112] Problem 3

problem number 1112

Added March 9, 2019.

Problem Chapter 4.5.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

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

Mathematica ✓

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

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

Maple ✓

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

$$w(x,y)=\mathit{\text{\_F1}}(-ax+y){\mathrm{e}}^{{\int }_{}^{x}b\ln {\left(\mathit{\text{\_a}}\lambda \right)}^k\ln {(-((-\mathit{\text{\_a}}+x)a-y)\beta )}^{n}d\mathit{\text{\_a}}}$$

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7.4.14.4 [1113] Problem 4

problem number 1113

Added March 9, 2019.

Problem Chapter 4.5.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$w_x+(ay+b{x}^n)w_y=c{\ln }^k(\lambda x)w$$

Mathematica ✓

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

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

Maple ✓

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

$$w(x,y)=\mathit{\text{\_F1}}\left(\frac{(-b{x}^n{\left(ax\right)}^{-\frac{n}{2}}\mathrm{WhittakerM}(\frac{n}{2},\frac{n}{2}+\frac{1}{2},ax){\mathrm{e}}^{\frac{ax}{2}}+(n+1)ay){\mathrm{e}}^{-ax}}{(n+1)a}\right){\mathrm{e}}^{\int c\ln {\left(\lambda x\right)}^{k}dx}$$

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7.4.14.5 [1114] Problem 5

problem number 1114

Added March 9, 2019.

Problem Chapter 4.5.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$ax{w}_x+by{w}_y=x^k(n\ln x+m\ln y)w$$

Mathematica ✓

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

$$\left\{\{w(x,y)\to c_1\left(y{x}^{-\frac{b}{a}}\right)\exp \left(\frac{x^k(akm\log (y)+akn\log (x)-an-bm)}{a^2{k}^2}\right)\}\right\}$$

Maple ✓

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

$$w(x,y)=x^{\frac{n{x}^k}{ak}}{\left(x^{\frac{b}{a}}\right)}^{\frac{m{x}^k}{ak}}{\left(y{x}^{-\frac{b}{a}}\right)}^{\frac{m{x}^k}{ak}}\mathit{\text{\_F1}}\left(y{x}^{-\frac{b}{a}}\right){\mathrm{e}}^{-\frac{(i\pi akm\mathrm{csgn}{\left(iy\right)}^3-i\pi akm\mathrm{csgn}{\left(iy\right)}^2\mathrm{csgn}\left(i{x}^{\frac{b}{a}}\right)-i(\mathrm{csgn}\left(iy\right)-\mathrm{csgn}\left(i{x}^{\frac{b}{a}}\right))\pi akm\mathrm{csgn}\left(iy\right)\mathrm{csgn}\left(iy{x}^{-\frac{b}{a}}\right)+2an+2bm)x^k}{2a^2{k}^2}}$$

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7.4.14.6 [1115] Problem 6

problem number 1115

Added March 9, 2019.

Problem Chapter 4.5.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

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

Mathematica ✓

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

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

Maple ✓

restart; 
pde :=  a*x^k*diff(w(x,y),x)+ b*y^n*diff(w(x,y),y) = (c*ln(lambda*x)^m+s*ln(beta*y)^t)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

$$w(x,y)=\mathit{\text{\_F1}}\left(\frac{(k-1)a{y}^{-n+1}-(n-1)b{x}^{-k+1}}{(k-1)a}\right){\mathrm{e}}^{{\int }_{}^x\frac{(c\ln {\left(\mathit{\text{\_a}}\lambda \right)}^m+s\ln {\left(\beta {\left(\frac{(k-1)a{y}^{-n+1}+(n-1)b{\mathit{\text{\_a}}}^{-k+1}-(n-1)b{x}^{-k+1}}{(k-1)a}\right)}^{-\frac{1}{n-1}}\right)}^t){\mathit{\text{\_a}}}^{-k}}{a}d\mathit{\text{\_a}}}$$

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