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Calculus and Beyond Homework Help
The Dirac Delta Function
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[QUOTE="Summer95, post: 5491253, member: 527464"] [h2]Homework Statement [/h2] Differential equation: ##Ay''+By'+Cy=f(t)## with ##y_{0}=y'_{0}=0## Write the solution as a convolution (##a \neq b##). Let ##f(t)= n## for ##t_{0} < t < t_{0}+\frac{1}{n}##. Find y and then let ##n \rightarrow \infty##. [SIZE=4]Then solve the differential equation with ##f(t)=\delta(t-t_{0})##.[/SIZE] [h2]Homework Equations[/h2] [SIZE=4][/SIZE] Convolution (Boas) Laplace Transforms (Boas) [h2]The Attempt at a Solution[/h2] So when I go through the first part with ##f(t)= n## for ##t_{0} < t < t_{0}+\frac{1}{n}## and do convolution I get ##y=\frac{1}{A(b-a)}\int_0^t(e^{-a(t-\tau)}-e^{-b(t-\tau)})f(\tau)d\tau## which has different cases depending on t: 0 if ##t<t_{0}## ##\frac{n}{A(b-a)}(\frac{1}{a}(1-e^{a(t_{0}-t)})-\frac{1}{b}(1-e^{b(t_{0}-t)}))## if ##t_{0}<t<t_{0}+\frac{1}{n}## ##\frac{n}{A(b-a)}(\frac{1}{a}(e^{-a(t-t_{0}-\frac{1}{n})}-e^{a(t_{0}-t)})-\frac{1}{b}(e^{-b(t-t_{0}-\frac{1}{n})}-e^{b(t_{0}-t)}))## if ##t>t_{0}+\frac{1}{n}## I don't understand what happens as ##n\rightarrow \infty##. I know it should become ##\frac{1}{A(b-a)}(e^{a(t_{0}-t)}-e^{b(t_{0}-t)})## for ##t>t_{0}## because that is what I get when I use ##f(t)=\delta(t-t_{0})## from the beginning. But what happens to all the extra terms? And the n out front that goes to infinity? [/QUOTE]
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The Dirac Delta Function
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