Question

I have a question about the unit step function in a differential equation. Here is a link to the "quiz" this comes from: http://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-iii-fourier-series-and-laplace-transform/unit-step-and-unit-impulse-response/MIT18_03SCF11_s25_3quizq.pdf

I will post the result I obtained in a followup post where I can actually use LaTeX.

Answer 1

For v(t), I obtained:
\[ce^{-kt}, t < 0; ce^{-kt}+\frac{1}{k}, t > 0.\]
With the given initial condition, c = 0, and v(t) becomes:
\[0, t < 0; \frac{1}{k}, t > 0.\]

Then the derivative of that is delta(t), is it not? Then delta(0+) = 0.

However, the soln given is delta(0+) = 1.
http://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-iii-fourier-series-and-laplace-transform/unit-step-and-unit-impulse-response/MIT18_03SCF11_s25_3quiza.pdf

Answer 2

It looks like where I went wrong was in assuming c to be the same for both parts of the piecewise solution. What I should have had for v(t) is:\[c_1e^{-kt}, t < 0; c_2e^{-kt}+\frac{1}{k}, t > 0.\]such that \[c_1 = 0; c_2 = \frac{-1}{k}\]