Differential equation for motion in mass-spring system with impulse

[zpeters1]

I am reviewing for a final and I don't know how an impulse affects the differential equation for motion in this mass-spring system. Can someone please help?

A mass m=1 is attached to a spring with constant k=2 and damping constant c=2. x(0)=0 & x'(0)=0. At the instant t=π, the mass is struck with a hammer, providing an impulse p=10.

Write the differential equation governing the motion of the mass.

 

[HallsofIvy]

Any external force applied to the object like that adds an additional function on the right side. You have a non-homogeneous equation with the impulse (delta) function being the additional function.

 

[shayaan_musta]

Do you know the form of homogenous differential equation?

ay''+by'+cy=0

Now as impulse is also given in your question. It means it is an additional quantity. The question can be solve without this aid.
Then you use your homogenous equation of the form given above.
It implies that any additional quantity adds a function in the homogenous equation and make it a non homogenous equation as,

ay''+by'+cy=g(x)

First you try to write your homogenous equation according to your question.