Ordinary Differential Equations

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Homework Statement



Give the general solution to the IVP

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

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

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.
 
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Answers and Replies

  • #2
vela
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Integrate the differential equation from [itex]\tau-\varepsilon[/itex] to [itex]\tau+\varepsilon[/itex] and take the limit as [itex]\varepsilon\rightarrow 0[/itex]. That will give you a result that tells you how big the discontinuity in y(t) is at [itex]t=\tau[/itex].
 
  • #3
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Integrate the differential equation from [itex]\tau-\varepsilon[/itex] to [itex]\tau+\varepsilon[/itex] and take the limit as [itex]\varepsilon\rightarrow 0[/itex]. That will give you a result that tells you how big the discontinuity in y(t) is at [itex]t=\tau[/itex].

So what would we get when we integrate [tex]\delta[/tex](t-[tex]\tau[/tex])? would it be 1?
 
  • #4
vela
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Yes, because the interval of integration includes the point [itex]t=\tau[/itex].
 

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