2.1.66 \(u_x+u u_y=1\) with \(u(0,y)=a y\). Problem 3.16(a) Lokenath Debnath

problem number 66

Added June 3, 2019.

Problem 3.16(a) nonlinear pde’s by Lokenath Debnath, 3rd edition.

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

\[ u_x+u u_y=1 \]

With \(u(0,y)=a y\).

Mathematica ✓

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

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

Maple ✓

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

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

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