2.1.63 \(x u_x+y u_y=x^2+y^2\) with \(u(x,1)=x^2\). Problem 3.14(e) Lokenath Debnath

problem number 63

Added June 3, 2019.

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

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

\[ x u_x+y u_y=x^2+y^2 \]

With \(u(x,1)=x^2\) with \(x>0,y>0\).

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x*D[u[x, y], x] +y*D[u[x, y], y]== x^2+y^2; 
 ic=u[x,1]==x^2; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[{pde,ic} ,u[x, y], {x, y},Assumptions->{x>0,y>0}], 60*10]];
 

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

Maple ✓

restart; 
pde :=x*diff(u(x,y),x) + y*diff(u(x,y),y)= x^2+y^2; 
ic  := u(x,1)=x^2; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde,ic],u(x,y)) assuming x>0,y>0),output='realtime'));
 

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

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