Category: Course Activities

Day 25 In-Class Group Work – Laplace and Inverse Laplace Transforms

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

Numerical Methods Calculator Examples

Numerical Methods Calculators can be built in any programming language or other tool capable of basic computation. The following examples all compute the answer to this exercise (also covered in class):

Exercise. Given the initial value problem $y’ = 3 – 2x – 0.5y, y(0)=1$, approximate the value of $y(1)$ using Euler’s Method with 10 intervals.

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Numerical Methods Project

Numerical methods provide a way to compute (approximate) values of solutions to differential equations, even when we cannot solve the equations exactly.  The drawback is the large number of numerical calculations required to obtain a desired value and level of precision.  In this project, you will use technology to implement the various numerical methods and use your technological solution to solve differential equations problems.

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