This is just asking for the ODEs and not to solve the entire problem. For example,
∂^2u/∂x^2 + ∂^2u/∂y^2 = 0 implies u(x,y) = φ (x)h(y) yields h d^2 φ /dx^2 + φ d^2h/dy^2 = 0.
Then dividing by φ (h) yields 1/ φ d^2φ/dx^2 = - 1/h d^2h/dy^2 = -λ
Or d^2φ/dx^2= -λ φ and d^2h/dy^2 = λh
I know. But you need to know the boundary conditions are homogenous, watch why we need that.
I am assuming that are numbers.∂u/∂t = k ∂^2u/∂x^2 – v_0 ∂u/∂x
Let then,
.
Divide by and we get,
Now the LHS only has while RHS only has thus,
for some real number .
This means that,
.
As far as seperation of variables goes that is the seperated equation.
But if we want to continue we need to determine what is. That is we need to determine those values of such that those two equations have non-trivial solutions. This is done by using boundary conditions, thus if they are homogeneous then . But it seems you do not have to go that far. So I stop here.