Why do you add independent solutions when finding a general equation for SHM?

Question

In my physics class, we have an assignment based on simple harmonic motion with the differential equation:
$$
frac{d^2x}{dt^2} + afrac{dx}{dt} + a^2x = 0
$$
Different parts of the question help us find two independent solutions of the form $x = Ae^{-Bt}$ and $x = Cte^{-Dt}$.
We are then given some information so that we can find $A$ and $C$ if $B = D = a$.
Our lectures tell us that we can find a general solution by adding the two independent solutions so that we get $x = (A + Ct)e^{-at}$ but I do not for the life of me understand why.
Or maybe I just misunderstood the lectures. But I want to understand either way.

J. Murray · Answer

Well, you've already found that $x_1(t) = e^{-at}$ and $x_2(t) = t e^{-at}$ are solutions to the differential equation.  It's not hard to show (though you should, as an important exercise) that any linear combination of $x_1$ and $x_2$ is also a solution, i.e. $y(t) = c_1 x_1(t) + c_2 x_2(t)$ also obeys the equation $y''(t) + ay'(t) + a^2 y(t)=0$.
The leap of faith that you're being asked to take is the converse - that every solution to that equation can be expressed as some linear combination of $x_1$ and $x_2$.  Therefore, to find the most general possible solution we take $y(t) = c_1 x_1(t) + c_2 x_2(t)$ and then apply initial conditions (or boundary conditions) to figure out what $c_1$ and $c_2$ should be in any specific case.

Why do you add independent solutions when finding a general equation for SHM?

One Answer

Add your own answers!

Ask a Question