The problem and my work is attached.
Vin is initially -5V for a long time before it switches to 5V at t=0.

From that, I wrote down the initial conditions by having Vin = -5, replacing the inductor with a short and the capacitor with an open. That's how I found Vc(0) and iL(0).

Vc forced response is 0.

After writing down those two equations in the image, I got stuck. I'm pretty sure I made a mistake somewhere because K1 would be 0 which would make my complete equation 0.

 

Why would K1 be 0?

Be specific. I think I know what you were thinking and where you went wrong, but I can't know for sure until you show enough work to confirm/refute my suspicion.

What are the limits of integration in your last equation (the one that you meant to say was equal to i_L(t=0+) and not i_C(t=0+))?

What does it mean to integrate something between 0 and 0+?

Perhaps you should write i_L in terms of the other KCL terms.

 

Why would K1 be 0?

Be specific. I think I know what you were thinking and where you went wrong, but I can't know for sure until you show enough work to confirm/refute my suspicion.

What are the limits of integration in your last equation (the one that you meant to say was equal to i_L(t=0+) and not i_C(t=0+))?

What does it mean to integrate something between 0 and 0+?

Perhaps you should write i_L in terms of the other KCL terms.

I tried to write i_L in different terms. Its in an image in the attachments. The first equation I used node equations but it looks like another differential equation, so I'm not sure if this is the correct direction. The second one didn't help me either.

 

The first one should work. Now express it in terms of your solution.

Also, start tracking your units!

 

But check your equation closely. I think you have a sign error.