How Does Changing Spring Constants Affect Mass Oscillation?

[wripples]

Homework Statement

So the problem is that you have a mass m connected to two springs, where each of the two springs is connected to a wall, as such:

|--------M--------|

Each spring first has a spring constant k and a equilibrium length L, and then when set up in the above configuration, M is 2L away from both walls (so both springs are stretched a length L from their equilibrium). Then the spring on the right has its spring constant instantaneously changed to 3k (instead of k), and the goal is to find the resulting function to describe the motion of M. We are to take its initial position to be 0 (so x(0) = 0).

2. The attempt at a solution

So my approach is to first set up the problem. First, take positive x to be displacement to the right. We have x(0) = 0, and since the right spring constant is supposed to instantaneously change to 3k, I assumed v(0) = 0. On the other hand, I'm not entirely sure that is right; perhaps I should solve for the acceleration at time t = 0 using the fact that both springs are stretched L past their equilibrium?

Afterwards, I try to find the net force. On the left, the spring with spring constant k has restoring force F_1 = -k(L+x), since it is already displaced by L to begin with, so it should already be acting with a force of -kL when we are at time t = 0, and stretching to the right (positive x) should increase the restoring force. Similarly, the force of the spring on the right is F_2 = -3k(-L+x); at the start, it should already have a force of magnitude 3kL pointing to the right, so it makes sense for x = 0, and as x increases (moves right), its restoring force should decrease.

Now, equating with F_2 - F_1 = F_net = ma, we have ma = 2kL - 4kx, which seems like its a particularly nasty differential equation to solve, due to the inhomogeneous term 2kL. Perhaps this is correct? And if so, how does one such solve a differential equation? I'm particularly rusty on doing that type of stuff.

Thanks!

 

[nasu]

I suggest to calculate everything relative to the new equilibrium position.
You can find this position follows from F=0 (or a=0)
Then you know the solution for motion around the equilibrium. The initial position is L/2 from equilibrium (if your formula for acceleration is right) so the amplitude will be L/2. Find the frequency from the new net force. Use initial conditions to find initial phase.
Then you can add L/2 in order to shift back the origin in the middle.