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

problem number 45

Added June 3, 2019.

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

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

\[ y u_x+x u_y=0 \]

with \(u(0,y)=e^{-y^2}\)

Mathematica ✓

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

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

Maple ✓

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

\[u \left (x , y\right ) = {\mathrm e}^{\left (x -y \right ) \left (x +y \right )}\]

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