# Help solving 2nd order differential equation

Help solving 2nd order differential equation plz

While solving for the time it takes an object of mass, m, with initial velocity, v, to compress a spring with spring constant, k to the maximum compression, I came accross the following differential equation

m(d$$^{2}$$x/dt$$^{2}$$) = kx - mg

I drew a f.b.d. of the forces on the object, mg down and kx (force of spring) up, and that's why I got kx - mg as the net force, the spring is on the bottom.

Can someone show me the technique to solving this please? I'm in grade 12 and have never learned differential equations before, but I finished both integration and differentiation calculus on my own in the summer.

Isn't this just a second order homogeneous linear ODE?

mx'' - kx = -mg
(-1/g)x'' + (k/mg)x = 0

With auxiliary polynomial:

(-1/g)r2 + k/mg = 0

So just solve for r and the answer is of the form:

$$x(t)=c_1e^{r_1t}+c_2e^{r_2t}$$

where c_1 and c_2 are constants.

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Thanks i didnt see that!

Wait a min...-mg/-mg = 1 not 0...

Wait a min...-mg/-mg = 1 not 0...

Yes, sorry. My excuse is that I just woke up when I originally replied. I replied to your private message with an explanation of how to find the (correct) solution.

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Yes this helped a lot! =D Thanks! ps sry it took me so long to get back, but i've read the solution right after you mailed it to me and i hadnt have time to respond