2.1.40 \((y+u) u_x+y u_y=x-y\) Problem 3.3(f) Lokenath Debnath

problem number 40

Added June 3, 2019.

Problem 3.3(f) nonlinear pde’s by Lokenath Debnath, 3rd edition.

Solve for \(u(x,y)\) \[ (y+u) u_x+y u_y=x-y \]

Mathematica

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

\[\left \{\left \{u(x,y)\to -\frac {\sqrt {c_1{}^2 x^2+2 c_1 x y+y^2+2 c_1 c_2}+y+c_1 y}{c_1}\right \}\right \}\]

Maple

restart; 
pde :=(y+u(x,y))*diff(u(x,y),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'));
 

\[u \left (x , y\right ) = \frac {-c_{1} y -y +\sqrt {c_{1}^{2} x^{2}+y^{2}+c_{1} \left (2 x y +2 c_{2}\right )}}{c_{1}}\]

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