I need to find a solution for the following PDE:$$ \begin{cases} y u_x - x u_y = 0 \\ u(x, 0) = x^2 & ,x \in \mathbb{R} \\ \end{cases}$$
My attempt:
I started by using the Lagrange method:$$A(x,y,u)u_x - B(x,y,u)u_y = C(x,y,u)$$$$\frac{dx}{A(x,y,u)} = \frac{dy}{B(x,y,u)} = \frac{du}{C(x,y,u)} $$
Therefore:$$\frac{dx}{y} = \frac{dy}{-x} \space\space\space \setminus \cdot(-xy) \\\\-x dx = y dy \space\space\space \setminus \int \\\\-\frac{x^2}{2} = \frac{y^2}{2} + C$$
Which simplifies to:$$x^2 + y^2 = C_1$$
Based on this characteristic equation and the face that $C(x,y,u)=0$, I proposed that the solution $ u $ can be expressed as a function of $ x^2 + y^2 $:
$$ u(x, y) = F(x^2 + y^2) $$
Given the initial condition, I deduced:
$$ u(x, 0) = F(x^2) = x^2 $$
This implies:
$$ F(t) = t $$
Final solution is:
$$ u(x, y) = x^2 + y^2 $$
Questions:
- Verification: At the beginning of the solution I divided by $x,y$. Does my solution $ u(x, y) = x^2 + y^2 $ correctly solve the PDE across the entire plane, including the origin where $ x = 0 $ or $ y = 0 $?
- Method of Characteristics: How should I handle the division by zero that arises when using the Lagrange method? Specifically, what approach should be taken when $ x = 0 $ or $ y = 0 $ to ensure the solution is valid everywhere?
I am seeking feedback on the correctness of my approach and any suggestions on how to address potential issues with the method used, particularly concerning divisions by zero.