Question

Find the LaPlace transforms for the following functions

(a) f(t)=3t^2-sin3t

(b) $\begin{bmatrix} t+1 &0\leq t\leq 1 \\ 2-e^{^-t} &t\geq 1 \end{bmatrix}$

Answer 1

Note: as The unit step function is defined ult-a) = lo if tca if tya at

(ID) Note: <{ ult- a)} = eas If <q gets = G(3), then aquct-a) q1t -a) = 2008 65) <{thy and akent = 3 - 1 ght1 (IV) ay sin (btty 3² +6² ni (III) against b jta

f(t) = 31² 31² - sin(3) Then {f(t) = 24312 - Bin (37)} = 32{12} - 24 sin (37)} 2! 3 2² +22 ! 32+1 10 3 2 2 +9 54 +632-393 - 3682252-18) 13 (32 +9) 23 (92 +9)

t +1 f(t) = · 0<t <1 2 - et , E21 1) + Then f(t) = (t + 1) [U (t - 0) – ult - 1) (2-ét).ult - 1)

= f(t) = (x + 1) ult-o) + [-+ +1-64] ult-1) + f(t) = (+1) ult - o) +[-(4-1) - 544=] uct -1) Therefore 24 fets <<lt +1) ult-0) + [-(4-1) - Just-1)} <<<t + )u(+ - 0} + <[-17 - -) - = pod kht + 27 + et ah-t-ety -(4-1) こ e -(t-1) 1) Just -1)}

= 1. [<1ty + x{1}] + e[-2{ty - 12 & {e ty] = (53+ $) + és [- - Bull - 1-es 32 + -3-1 l 3+1 et