113k views
1 vote
DEFINITION 7.1.1 Laplace Transform Let f be a function defined for t ≥ 0. Then the integral ℒ{f(t)} = [infinity] e−stf(t) dt 0 is said to be the Laplace transform of f, provided that the integral converges. to find ℒ{f(t)}. (Write your answer as a function of s.) f(t) = cos(t), 0 ≤ t < π 0, t ≥ π ℒ{f(t)} = (s > 0)

1 Answer

6 votes


f(t)=\begin{cases}\cos t&amp;\text{for }0\le t<\pi\\0&amp;\text{for }t\ge\pi\end{cases}

Write
f(t) in terms of the step function
u(t):


f(t)=(u(t)-u(t-\pi))\cos t

where the step function is defined by


u(t)=\begin{cases}1&amp;\text{for }t\ge0\\0&amp;\text{for }t<0\end{cases}

The Laplace transform is then


\mathcal L_s\{f(t)\}=F(s)=\displaystyle\int_0^\infty f(t)e^(-st)\,\mathrm dt=\int_0^\pi\cos t\,e^(-st)\,\mathrm dt


\implies F(s)=((e^(-\pi s)+1)s)/(s^2+1)

(if you're stuck on the integral, try integrating by parts)

User Yofee
by
8.5k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories