2.1.51 \(x u_x+y u_y=u+1\) with \(u=x^2\) on \(y=x^2\) Problem 3.5(g) Lokenath Debnath

problem number 51

Added June 3, 2019.

Problem 3.5(g) nonlinear pde’s by Lokenath Debnath, 3rd edition.

Solve for \(u(x,y)\) \[ x u_x+y u_y=u+1 \] with \(u=x^2\) on \(y=x^2\).

Mathematica

ClearAll["Global`*"]; 
pde =  x*D[u[x, y], x] + y*D[u[x,y],y] == u[x,y]+1; 
ic  = u[x,x^2]==x^2; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[{pde,ic} ,u[x, y], {x, y}], 60*10]];
 

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

Maple

restart; 
pde :=x*diff(u(x,y),x)+ y*diff(u(x,y),y)= u(x,y)+1; 
ic  := u(x,x^2)=x^2; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y)) ),output='realtime'));
 

\[u \left ( x,y \right ) ={\it \_F1} \left ( {\frac {y}{x}} \right ) x-1\] Maple does not accept this form of Cauchy data as I.C.

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