Using the definition of Laplace transform and other theorems you find, determine: L{t(integral {0, t}) e^(-3u) du} Identify each theorem as you use it.
If someone would help me with this I would really appreciate it. This problem will be on the final, so if you could clearly state which theorems need to be used for each step, I would really appreciate it.
You want the LT of:
f(t) = t [integral_{0, t}) e^(-3u) du
differentiate twice to eliminate the integral (we use the product rule and the
fundamental theorem of calculus:
f'(t) = integral_{0,t} e^{-3u} du + t e^{-3t}
f''(u) = 2 e^{-3t} - 3 t e^{-3t)
So Lf''(s) can be evaluated by table look-up, call it G(s), then:
Lf''(s) = s^2 F(s) - s f(0) - f'(0)
where F(s) = Lf(s), so:
F(s) = [G(s) + s f(0) + f'(0)]/s^2
Now f(0) = 0, and f'(0) = 0 so:
F(s) = G(s)/s^2
RonL