Differential equations of a forced coupled spring-pendulum system

Currently working on a problem and I can really figure out how to write the differential equations for it. Here’s the situation:

So we have a mass $m$ tied to the wall with a spring of constant $k$. The wall itself is oscillating and its position is given by $x_0 cos(omega t)$. Tied to the mass is a pendulum of length $L$ and mass $m$. I have to figure out the equation for the amplitude. My guess so far is the following (1 being the spring-mass and 2 being the pendulum):

$$m_1ddot{x}_1=-k(x_1-x_0cos(omega t))+frac{mg}{L}(x_2-x_1)$$
$$m_2ddot{x}_2=frac{-mg}{L}(x_2-x_1)$$

But I’m not entirely sure about them. Since the pendulum is tied to the moving mass, I’ve considered writing the second one as:

$$m_2(x_2-x_1)”=frac{-mg}{L}(x_2-x_1)$$

But I’m not sure the logic holds. If that’s of any help, I have to prove that the equations for the amplitude are:

$$A_1=frac{kx_0(g-Lomega^2)}{mLomega^4-(2mg+kL)omega^2+kg}$$
$$A_2=frac{kgx_0}{mLomega^4-(2mg+kL)omega^2+kg}$$

EDIT:

It seems the first differential equations were right, I managed to obtain the correct amplitudes by substituting the complex solutions $x_1 rightarrow z_1=A_1e^{iomega t}$ and $x_2 rightarrow z_2=A_2e^{iomega t}$. Leaving this here in case it could help someone.