Lesson 26: Inverse Laplace Transform

Hi everyone! Read through the material below and watch the videos, then give the WeBWorK assignment a try.

Review: Finding the Laplace Transform of a function

You may wish to refer to this Table of Laplace Transforms during today’s lesson.

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

That’s it for today! Try out the WeBWorK assignment, and let your professor know if you have questions.