I got an exercise in my PDE class which I'm struggling to solve.
Solve following eq using the method of characteristics $$u_x(x,y)+xu_y(x,y) = 1 \qquad (x,y) \in \mathbb{R}^2$$ $$u(3,y) = y^2 \qquad y \in \mathbb{R}$$
My approach was :
To find characteristics solve $(x'(t),y'(t)) = (1,x(t)) $
So I got $\quad x(t) = t+x_o; \quad y(t) = \frac{1}{2}t^2+x_0t+y_0 $
Now we want our characteristics to start at a curve where we know the value of $u$, hence start at $\Gamma = \{ (3,s) : s \in \mathbb{R} \}$.
We get $\quad x_0 = 3 ; \quad y_0 = s$
Now $u'(x(t),y(t)) = u_x(x(t),y(t)) + x(t)u_y(x(t),y(t)) = 1$ hence $u(x(t),y(t)) = t+ u_0 $ where $u_0 = s^2$
So we get $u((t+3),(\frac{1}{2}t^2+3t+s)) = s^2 +t$
I couldn't find an easy way to calculate the equation for $u$. This is the point where I started wondering, if everything was alright.
My approach to solve this would be using polynomial division but I think thats not the point of the exercise.