Variable Coefficient EquationsExpected Educational ResultsVariable Coefficient EquationsMethod of Solving Homogeneous Cauchy-Euler EquationsInvestigation 03Investigation 04Investigation 05CC 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 .

From

**Activity 02**, the characteristic equation is .Find the solutions to the characteristic equation, and .

- If , then the linearly independent solutions are and and the homogeneous solution is
- If , then the linearly independent solutions are and and the homogeneous solution is
- If and are complex numbers in the form , then the linearly independent solutions are and and the homogeneous solution is

- For what values of is
**not**a solution to ? Explain. - For what values of is a solution to ? Explain.
- Suppose is a solution to . Find the associated characteristic equation.

**Example 01**: Solve ,

**Solution**:

Note that the coefficients of each -term is a polynomial function of the independent variable in which the degree of the polynomial term is equivalent to the degree of the derivative of . Thus, this ode is a Cauchy-Euler equation.

Identify the coefficients from the ode: , , . and .

The, characteristic equation for the above Cauchy-Euler equation is: . Using *Mathematica*, the roots of the characteristic equation are: , where and .

Thus, the homogeneous solution is .

- Use the Wronskian to verify and , where , are linearly independent. Explain.
- Why is the condition, , above necessary? Explain.
- Use the Wronskian to verify and are linearly independent. Explain.

Solve the following homogeneous DEs:

- , .
- , .
- , .
- , .
- , .
*Hint*: Let .

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**Last Modified**: Wednesday, 14 October 2020 11:42 EDT