Question

Diff EQ. Laplace of piece wise function. I have the answer but I need the steps to solve explained. Look at attachment.

Answer 1

from first principles.....

\(L\{t ~ u(t-a) \} = \int_0^a 0 * e^{-st} ~ dt + \int_a^{\infty} t e^{-st} ~ dt\)

by IBP \(u = t, u' = 1, v' = e^{-st}, v = -{1\over s} e^{-st}\)

\(L\{t u(t-a) \} = 0 ~ + \left( - t {1\over s} e^{-st} \right)_a^{\infty} + \int_a^{\infty} {1\over s} e^{-st} dt\)

\(= \left( 0 + a {1\over s} e^{-as} \right) + \left( - t {1\over s^2} e^{-st} \right)_a^{\infty} \)


\(= a {1\over s} e^{-as} + \left( 0 + a {1\over s^2} e^{-as} \right) \)

\( = a e^{-as} ~ \dfrac{s + 1}{s^2} \)


hope there are no typoes in there, i latexed in a rush...

iam guessing you wanted to see it this way, or are you trying to use a table of transforms??!!