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 $t=\tau$.