2.1.59 \(u_x+x u_y=y\) with \(u(1,y)=2 y\) Problem 3.9(b) Lokenath Debnath

problem number 59

Added June 3, 2019.

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

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

Mathematica ✓

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

\[\left \{\left \{u(x,y)\to -\frac {x^3}{3}-\frac {x^2}{2}+x y+y+\frac {5}{6}\right \}\right \}\]

Maple ✓

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

\[u \left ( x,y \right ) =-{\frac {{x}^{3}}{3}}+yx+{\frac {5}{6}}-{\frac {{x}^{2}}{2}}+y\]

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