2.1.49 \(u_x+x u_y=(y-\frac {1}{2}x^2)^2\) with \(u(0,y)=e^{y}\). Problem 3.5(f) Lokenath Debnath

problem number 49

Added June 3, 2019.

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

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

\[ u_x+x u_y=(y-\frac {x^2}{2})^2 \]

with \(u(0,y)=e^{y}\).

Mathematica ✓

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

\[\left \{\left \{u(x,y)\to \frac {1}{4} x \left (x^2-2 y\right )^2+e^{y-\frac {x^2}{2}}\right \}\right \}\]

Maple ✓

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

\[u \left (x , y\right ) = x \,y^{2}-y \,x^{3}+\frac {x^{5}}{4}+{\mathrm e}^{y -\frac {x^{2}}{2}}\]

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