i have a differential equation.
∂h/∂t + [g sin a h²/v]∂h/∂x = 0. where h = h(x,t).
i need to show by substitution that the (implicit) general solution for h is h = f(x - (g sin a h²t/v)) where f is an arbitrary differential function of a single variable.
The Attempt at a Solution
∂h/∂x = ∂f/∂x
∂h/∂t = ∂f/∂t * (-g sin a h²/v)
subbing in gives
∂f/∂t*(-g sin a h²/v) + (g sin a h²/v)*∂f/∂x = 0
cancelling g sin a h²/v
-∂f/∂t + ∂f/∂x = 0
so now i have to show... ∂f/∂t = ∂f/∂x ???
seems like i'm going in circles...
any help will be appreciated