Question

Solve the Cauchy problem equation:

u_x + u_y = u^2, u(x,0) = 1.

Answer 1

u_x + u_y = u^2, u(x,0) = 1.
Method of characteristics:
dx = dy = du/u^2

1) du/u^2 = dx
Integrate:
-1/u + C1=x => C1 = 1/u + x

2) dx = dy
Integrate:
y + C2=x => C2 = x - y

Therefore, the complete integral of the equation:
F ( 1/u + x, x - y) = 0 or in another form:
u = ( f (x- y) - x )^-1 where f - some differentiable function.
The initial condition u(x,0) = 1.
u(x, 0) = ( f (x) - x )^-1 = 1 => f(x) = x+1
u(x,y) = 1/(x-y+1-x) = 1/(-y+1) = 1/(1-y)
Answer: u (x, y) = 1/(1-y)