2.1.68 $(y+u)u_x+(x+u)u_y=x+y$. Problem 3.17(a) Lokenath Debnath

problem number 68

Added June 3, 2019.

Problem 3.17(a) nonlinear pde’s by Lokenath Debnath, 3rd edition.

Solve for $u(x,y)$ $$(y+u)u_x+(x+u)u_y=x+y$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (y+u[x,y])*D[u[x, y], x] +(x+u[x,y])*D[u[x, y], y]== x+y; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde ,u[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=(y+u(x,y))*diff(u(x,y),x) + (x+u(x,y))*diff(u(x,y),y)=x+y; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y),'build') ),output='realtime'));
 

$$u(x,y)=\mathit{\_}\mathit{C}\mathit{1}x-x-\frac{1}{{(\mathit{\_}\mathit{C}\mathit{1}x+y)}^2}(\frac{{(-1/2+i/2\sqrt{3})}^6}{64{\mathit{\_}\mathit{C}\mathit{2}}^3}{(-2\frac{(\mathit{\_}\mathit{C}\mathit{1}x+y)\mathit{\_}\mathit{C}\mathit{2}}{\mathit{\_}\mathit{C}\mathit{1}+1}\sqrt[3]{(4+4\sqrt{\frac{4{(\mathit{\_}\mathit{C}\mathit{1}x+y)}^3{\mathit{\_}\mathit{C}\mathit{2}}^3+\mathit{\_}\mathit{C}\mathit{1}+1}{\mathit{\_}\mathit{C}\mathit{1}+1}}){(\mathit{\_}\mathit{C}\mathit{1}+1)}^2}+8{(\mathit{\_}\mathit{C}\mathit{1}x+y)}^2{\mathit{\_}\mathit{C}\mathit{2}}^2\frac{1}{\sqrt[3]{(4+4\sqrt{\frac{4{(\mathit{\_}\mathit{C}\mathit{1}x+y)}^3{\mathit{\_}\mathit{C}\mathit{2}}^3+\mathit{\_}\mathit{C}\mathit{1}+1}{\mathit{\_}\mathit{C}\mathit{1}+1}}){(\mathit{\_}\mathit{C}\mathit{1}+1)}^2}}+4i\sqrt{3}(1/2\frac{(\mathit{\_}\mathit{C}\mathit{1}x+y)\mathit{\_}\mathit{C}\mathit{2}}{\mathit{\_}\mathit{C}\mathit{1}+1}\sqrt[3]{(4+4\sqrt{\frac{4{(\mathit{\_}\mathit{C}\mathit{1}x+y)}^3{\mathit{\_}\mathit{C}\mathit{2}}^3+\mathit{\_}\mathit{C}\mathit{1}+1}{\mathit{\_}\mathit{C}\mathit{1}+1}}){(\mathit{\_}\mathit{C}\mathit{1}+1)}^2}+2{(\mathit{\_}\mathit{C}\mathit{1}x+y)}^2{\mathit{\_}\mathit{C}\mathit{2}}^2\frac{1}{\sqrt[3]{(4+4\sqrt{\frac{4{(\mathit{\_}\mathit{C}\mathit{1}x+y)}^3{\mathit{\_}\mathit{C}\mathit{2}}^3+\mathit{\_}\mathit{C}\mathit{1}+1}{\mathit{\_}\mathit{C}\mathit{1}+1}}){(\mathit{\_}\mathit{C}\mathit{1}+1)}^2}}))}^2+{(\mathit{\_}\mathit{C}\mathit{1}x+y)}^3)$$

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