2.1.47 \(u_x+x u_y=0\) with \(u(0,y)=\sin y\). Problem 3.5(d) Lokenath Debnath

problem number 47

Added June 3, 2019.

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

Solve for \(u(x,y)\) \[ u_x+x u_y=0 \] with \(u(0,y)=\sin y\).

Mathematica

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

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

Maple

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

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

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