# Characteristic base curves

$$u u_x + (y + 1) u_y = u , x ∈ R, y > 0$$ $$u(x,0)=-3x, x ∈ R$$

I found the solution of the above initial value problem to be $$u(x,y)= \frac{-3x(y+1)}{(-3y+1)}$$

If someone could confirm this it would be fantastic. How do I sketch the characteristic base curves and define the region where the solution can be determined?

The characteristic base curves are the straight lines $x = s(1−3y)$. Where these lines intersect at the point$(0, 1/3)$ .