Ordinary Differential Equations

[sassie]

Homework Statement

Give the general solution to the IVP

L[y]=y'+(sint)y=\delta(t-\tau)
y(0)=0

For all t>0 by placing a jump condition on y(t) and solving the differential equation for t<\tau and t>\tau

Homework Equations

The Attempt at a Solution

I'm plenty sure I can get the general solution to the problem, but I do not at all know how to get the "jump condition" as it wasn't explained in lectures or in the textbook. Your help is very much appreciated.

 

[vela]

Integrate the differential equation from \tau-\varepsilon to \tau+\varepsilon and take the limit as \varepsilon\rightarrow 0. That will give you a result that tells you how big the discontinuity in y(t) is at t=\tau.

 

[sassie]

Integrate the differential equation from \tau-\varepsilon to \tau+\varepsilon and take the limit as \varepsilon\rightarrow 0. That will give you a result that tells you how big the discontinuity in y(t) is at t=\tau.

So what would we get when we integrate \delta(t-\tau)? would it be 1?

 

[vela]

Yes, because the interval of integration includes the point t=\tau.