Where Does the δ(t) Term Come From in First Order Unit Step Response?

[charlie.elvers]

I think I mostly understand how this works, and it makes intuitive sense. However, I'm a little bit confused by one step in the proving of this.

On the second page of the attached PDF, there is this statement:

x' + kx = (r/k + c2)δ(t) ... (followed by the cases of t<0 and t>0

What comes after δ(t) is easy for me to follow, but I don't know where this δ(t) term comes from. Does this term evaluate to 0? If so, how? I would think it'd evaluate to (r/k + c2) because the (r/k + c2) is independent of t

Thanks.

 

[charlie.elvers]

Sorry folks. I just read the forum rules, and I think this is probably the wrong forum for such a thread.