Solving Initial Value ProblemsExpected Educational ResultsSolving Initial Value ProblemsMethodUse Technology to use the method of partial fractionsInvestigation 01CC BY-NC-SA 4.0

**Author**: John J Weber III, PhD
**Corresponding Textbook Sections**:

**Section 7.5**– Solving Initial Value Problems

**Objective 22–1**: I can use the Laplace transform to solve an initial value problem.

- Take the Laplace transform of both sides of IVP;
- Solve the transform for (more frequently referred to as );
- Take the inverse Laplace transform of both sides of the equation from step 2
*Recall*: .

**Example 01**:

Solve: , , .

**Solution**:

Apply Laplace transform:

Evaluate Laplace transforms:

Expand using algebra:

Substitute initial conditions:

Simplify/rewrite:

Factor out :

Rewrite:

Solve for :

Apply inverse Laplace transforms:

**Note**: Do **NOT** expand denominators.

Use method of partial fractions:

**Mathematica**

`1``(* Separate into partial fractions: 1/(s^2-5s+6)(1/(s^2+1)+1) *)`

2`Apart[1/(s^2-5s+6)(1/(s^2+1)+1)]`

**Warnings**:

- Be very
**careful**with the*syntax*.*Syntax*is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic*Mathematica*syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf. - To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

Use algebra to combine “like” terms:

Use knowledge of Laplace transforms:

Use the Laplace transform to solve the following IVPS:

- ,
- , ,
- , ,
- , ,
- , ,
- , ,
- , , ,

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**Last Modified**: Sunday, 8 November 2020 17:45 EDT