Solving DE Homework: g'(t) + ag(t) = \delta(t-\xi)

[squenshl]

Homework Statement

Solve, sketch & check it's accuracy by operating on a test function of g'(t) + ag(t) = $$\delta$$(t-$$\xi$$), g(t) = 0, t < $$\xi$$

Homework Equations

The Attempt at a Solution

I have solved it getting g(t) = c₁e^-at + e^a($$\xi$$-t)H(t-$$\xi$$) where H(x) is the Heaviside function. How do I sketch this & operate it on a test function.

 

[jackmell]

Homework Statement

Solve, sketch & check it's accuracy by operating on a test function of g'(t) + ag(t) = $$\delta$$(t-$$\xi$$), g(t) = 0, t < $$\xi$$

Homework Equations

The Attempt at a Solution

I have solved it getting g(t) = c₁e^-at + e^a($$\xi$$-t)H(t-$$\xi$$) where H(x) is the Heaviside function. How do I sketch this & operate it on a test function.

You can plot it manually if you wish. Just choose some values for a and e and c. But why not, if you're studying DEs, learn how to work with them in Mathematica. Here's the code I would use to "sketch" it and use a "test" function with the solution. Try and figure out what I'm doing if you're interested:

e = 5;
a = 1;
mysol = y /. DSolve[y'[t] + a y[t] == DiracDelta[t - e], y, t] // First
Plot[mysol[t] /. C[1] -> 1, {t, 0, 5}]

 

[squenshl]

I got the sketch, so do I investigate a test function in the usual way.

 

[jackmell]

I got the sketch, so do I investigate a test function in the usual way.

Hi. The act of solving it numerically implicitly generates a particular (test) solution to the DE. I mean you choose some values for a, e and the arbitrary integration constant, C[1], in my code, then run the numeric integrator NDSolve. That "creates" a numeric function which satisfies the DE.