Variable Coefficient EquationsExpected Educational ResultsVariable Coefficient EquationsDerivation of the Solutions to Cauchy-Euler EquationsActivity 02CC BY-NC-SA 4.0

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

**Section 4.7**– Variable Coefficient Equations

**Objective 14–1**: I can identify variable-coefficient equations.**Objective 14–2**: I can find a solution to a homogeneous Cauchy-Euler equation.**Objective 14–3**: I can find a solution to a nonhomogeneous Cauchy-Euler equation.**Objective 14–4**: Given a solution, I can apply the Reduction of Order Theorem to find another solution to a Cauchy-Euler equation.- Objective 14-5: I can find a solution to a Cauchy-Euler IVP.

**NOTE**: You will **not** be assessed on the derivation of the solutions to the Cauchy-Euler Equations.

Given the homogeneous Cauchy-Euler equation: , .

The **goal** is to rewrite the ODE as by making a substitution. *Explain why*.

Let .

Differentiate the

*let statement*with respect to .Rewrite equation from

**Step 2**using the*let statement*.Multiply the equation from

**Step 3**by .Consider the derivatives as differentials and simplify the side that contains product of the two derivatives.

Differentiate the equation from

**Step 5**with respect to .Using the equation from

**Step 6**:- Multiply the side with the single term by and consider the derivatives as differentials and simplify.
- Multiply the other side by .
- Explain why
**Step 7-1**and**Step 7-2**are valid.

Rewrite the side of the equation from

**Step 7**that has two terms using the result from**Step 5**.Rewrite as a second derivative the side of the equation from

**Step 8**that has a single term.Solve the equation from

**Step 9**for the second derivative that has a coefficient.Substitute the equations from

**Step 5**and**Step 10**into . Use**Step 1**to show . Now the DE is in terms of . Simplify.Suppose and are the roots to the characteristic equation for the simplified DE from

**Step 11**. Use the*Method of Undetermined Coefficients*to write the solutions to the DE.Use the

*let statement*to rewrite the solutions**Step 12**in terms of . Simplify. What do you notice?Suppose is a repeated root to the characteristic equation for the simplified DE from

**Step 11**. Repeat**Step 12**and**Step 13**. What do you notice?Suppose are complex roots to the characteristic equation for the simplified DE from

**Step 11**. Repeat**Step 12**and**Step 13**. What do you notice?

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**Last Modified**: Wednesday, 14 October 2020 11:40 EDT