2.1.58 \(u_x+x u_y=y\) with \(u(0,y)=y^2\) Problem 3.9(a) Lokenath Debnath

problem number 58

Added June 3, 2019.

Problem 3.9(a) nonlinear pde’s by Lokenath Debnath, 3rd edition.

Solve for \(u(x,y)\)

\[ u_x+x u_y=y \]

With \(u(0,y)=y^2\).

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[u[x, y], x] +x*D[u[x, y], y]== y; 
ic  = u[0,y]==y^2; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[{pde,ic} ,u[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{u(x,y)\to \frac {x^4}{4}-\frac {x^3}{3}-x^2 y+x y+y^2\right \}\right \}\]

Maple ✓

restart; 
pde := diff(u(x,y),x) + x*diff(u(x,y),y)= y; 
ic  := u(0,y)=y^2; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde,ic],u(x,y))),output='realtime'));
 

\[u \left (x , y\right ) = -\frac {x^{3}}{3}+x y +\left (-\frac {x^{2}}{2}+y \right )^{2}\]

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