2.1.65 \(x u_x+y u_y=x y\) with \(u=\frac {x^2}{2}\) at \(y=x\). Problem 3.14(g) Lokenath Debnath

problem number 65

Added June 3, 2019.

Problem 3.14(g) nonlinear pde’s by Lokenath Debnath, 3rd edition.

Solve for \(u(x,y)\) \[ x u_x+y u_y=x y \] With \(u=\frac {x^2}{2}\) at \(y=x\).

Mathematica

ClearAll["Global`*"]; 
pde =  x*D[u[x, y], x] +y*D[u[x, y], y]== x*y; 
 ic=u[x,x]==x^2/2; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde ,u[x, y], {x, y}], 60*10]];
 

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

Maple

restart; 
pde :=x*diff(u(x,y),x) + y*diff(u(x,y),y)= x*y; 
ic  := u(x,x)=x^2/2; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y)) ),output='realtime'));
 

\[u \left (x , y\right ) = \frac {x y}{2}+\textit {\_F1} \left (\frac {y}{x}\right )\]

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