1
Q
Seperation of variables
A
u(x,t) = X(x)T(t)
∂x^2 X(X) T(t) - X(x) ∂t^2 T(t) = 0
∂x^2 X(X) / X(x) = ∂t^2 T(t) / T(t) = provided constant
∂x^2 X(X) = const X(x)
∂t^2 T(t) = const T(t)
2
Q
General solutions of separation of variables
A
-k^2 x = Acos(kx) + Bsin(kx)
k^2 x = Aexp[kx] + Bexp[-kx]
k x = Aexp[kx]
3
Q
D’Alembert method
A
p = λx + y
A∂x^2u + B∂x∂yu + C∂y^2u = 0
∂x^2 = λ^2 d^2u/dp^2
∂y^2 = d^2u/dp^2
∂x∂y = λ d^2u/dp^2
substitute into the original eq and solve for λ
4
Q
∂x =
A
∂u/∂p ∂p/∂x
5
Q
∂x^2 =
A
∂x/∂x + 2∂x ∂^2u/∂p^2
6
Q
∂y
A
∂u/∂p ∂p/∂y
