# Simple Harmonic System Equations

Question:

Equations:

I'm having trouble understanding what a "solution" to equation 6 refers to? What are the implications of including gravity in the equation?

$$Mg$$ is a constant so only changes the equilibrium position, not the angular frequency. Equation 6 can be rearranged to give
$$\frac{d^2y}{dy^2}=-\frac{k}{M}(y+L_0+\frac{Mg}{k})$$
This makes the new equilibrium position $$L_0+\frac{Mg}{k}$$ .
Mathematically this is a differential equation. But, a solution is any function that you write down and replace $$y(t)$$ with $$guess(t)$$ and $$y(t)_{tt}$$ with $$guess(t)_{tt}$$, and the equality will still hold.
The implication of including gravity is mathematically making it $$F(y,y_{tt})=const \neq 0$$ instead of having a homogeneous and boring $$F(y,y_{tt})=0$$. In this case you just need to add it to your guessed or calculated solution to the homogenous equation and it will be true. And it will also be a general solution, so cheating works.