How Can Mathematical Modeling Describe Oscillations in a Weighted String System?

[Firben]

A string of length L are connected in x = 0 and x = L. The point x = a, 0<a<L, are a point formed weight with mass m₀ attached. Formulate the mathematical problem for small transversal oscillations .

I got it to be:
∴
m₀*u_tt(a,t) = s(a,t)(u_x(a⁺,t)-u_x(a^-,t))
u(x,0) = g(x)
u_t(x,0) = h(x)

The answer should be

∴
u_tt - c²u_xx = -(m₀*g/ρ)*δ(x-a)

How can i transform it ?

 

[kurt.physics]

We can transform it by using the chain rule:utt - c2uxx = -(m0/ρ)*g'(a)δ(x-a)where g'(a) is the derivative of g(x) evaluated at x = a.