What is the equations of motion for a pendulum and spring

[swooshfactory]

Homework Statement

This is for a math based physics class. I need the equation ofa pendulum and of a spring.

Homework Equations

Spring: x=Asin(wt-phi)+B
Pendulum: theta=theta0cos(wt+phi) or theta=theta0cos(sqrt(g/l)sin(theta))

The Attempt at a Solution

I don't know which of these are correct. Please let me know the right equation for each; if it's a differential equation, please let me know what it is solved if possible. Thank you.

 

[AEM]

For small displacements of the pendulum when $$sin \theta \simeq \theta$$ and ignoring friction in both cases, the equations governing the motion of the mass-spring and the pendulum are those of simple harmonic motion.

The D.E. is $$\frac {d^2 x}{ dt^2} + \omega^2 x = 0$$

Which has a solution with two arbitrary constants (necessary because the D.E. is second order). There are various equivalent forms for the solution. One of which is

$$x = A sin(\omega t + \phi)$$

$$\omega$$ is called the angular frequency and its exact form in terms of the physical parameters in your problem can be determined when you set up the differential
equation. A is the amplitude and $$\phi$$ the phase. These two parameters are determined from initial conditions. All of this is covered in standard physics textbook.