Choose one member of your group to be the scribe. The scribe should record your group’s work as you complete the examples below. You may wish to refer to this Table of Laplace Transforms when working on the problems below.

Example 1.  Find the Laplace Transform of each function, and determine the interval on which it is defined.

  1. $t^{2}+4 t^{3}-7 t^{6}$
  2. $\sin t+\cos 3 t$
  3. $5 e^{2 t}-4 \sin 3 t$
  4. $5 t^{2}+3 \sin 5 t-2 e^{6 t} \cos 2 t$

Find the inverse Laplace Transform

Now we consider the inverse problem – if I give you the Laplace Transform of a function, can you find the function it came from?

Example 2. Find the inverse Laplace Transform of each function.

  1. $\frac{3}{s}+\frac{4 !}{s^{5}}+\frac{1}{s-7}, \quad s>7$
  2. $\frac{1}{s^{2}+25}+\frac{s}{s^{2}+25}, \quad s>0$
  3. $\frac{1}{(s-4)^{2}+9}, \quad \mathrm{~s}>4$
  4. $\frac{1}{(s-6)^{7}}+\frac{5}{2 s-7}, s>6$
  5. $\frac{5}{s^{2}-8 s+41}, s>4$