Question

PDE help, please
Solve
\(tU_x - xU_t=0\\U(x,0)= 2x^2\)

Answer 1

\(-\dfrac{t}{x}U_x+U_t =0\)
Seek \((x(t),t)\) such that \(U(x(t),t)\) is a constant along it.
characteristics equation
\(\dfrac{\partial U}{\partial t}=\dfrac{\partial U}{\partial x}\dfrac{\partial x}{\partial t}+\dfrac{\partial U}{\partial t}=0\)
Make a comparison to the problem, I have
\(\dfrac{\partial x}{\partial t}=\dfrac{-t}{x}\)
Which gives me \(\dfrac{x^2}{2}=\dfrac{-t^2}{2}+ x_0 (constant)\)

Answer 2

Hence \(2x_0= x^2+t^2\)