Variable Coefficient EquationsExpected Educational ResultsVariable Coefficient EquationsReduction of OrderTheorem: Reduction of OrderExample 03:Method of Solving Nonhomogeneous Cauchy-Euler EquationsInvestigation 08Investigation 09CC 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.

Let be a solution to the homogeneous DE on some interval. Then, , where is the coefficient of the first-order derivative term, is a second, linearly independent solution.

Consider . Since the second-order derivative does **not** have a coefficient with , this is **not** a Cauchy-Euler equation.

Verify is a solution. Find .

**Solution**:

So, . Thus, is a solution to the ode.

Since .

Let's find . By the *Reduction of Order* definition:

Thus, .

There are two options:

- Use method of variation of parameters; or
- Find one solution and apply the Reduction of Order Theorem

Solve the following nonhomogeneous DEs:

- , [Hint: Use a substitution .]
- , .

Solve the following homogeneous DEs:

- Suppose is a solution to , . Find the general solution.
- Suppose is a solution to , . Find the general solution.

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**Last Modified**: Wednesday, 14 October 2020 2:48 EDT