Laplace Transforms: Unit Step Functions

1. Find the Laplace transform of the given function.  solution    

 (a)  \(f(t) = 2u_{2}(t)+3(t-5)u_5(t)-7(t-3)^4\,u_3(t)\) 

 (b)  \(f(t) = (t-1)u_2(t)-2(t-2)u_1(t)  \)

2. Find the Laplace transform of the given function.  solution

(a)  \(f(t)=\begin{cases} t^2,    \: \: 0 \leq t < 3\\ 1, \: \:\:  t \geq 3  \end{cases} \)  

(b) \(f(t)=\begin{cases} 1,    \: \: 0 \leq t <2 \\ t,  \:\: 2 \leq t < 5 \\ 0,   \:\:   t \geq 5  \end{cases} \) 

3. Find the Laplace transform of given function.    solution

  (a) \(f(t) = 2u(t-1)(t-1)^2+3\delta(t-5) \)  

  (b)  \(f(t) = 2u(t-1) t^2+5\delta(t-3) \)  

4. Find the Laplace transform of the following functions.  solution

 (a)  \(h(t) = e^{2t}*\sin (3t) \) 

 (b)  \(\displaystyle{h(t) = \int_0^t \cos (2v) \, dv }\)

  (c)  \(\displaystyle{h(t) = \int_0^t e^{t-\tau} \cosh (\tau) \, d\tau }\)