# Variable Coefficient Equations

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

• Section 4.7 – Variable Coefficient Equations

## Expected Educational Results

• 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.

## Variable Coefficient Equations

### Higher-Order Cauchy-Euler Equations

NOTE: You will not be assessed on solving third-order [or higher] Cauchy-Euler Equations.

#### Investigation 06

NOTE: You will need to derive the characteristic equation using a similar derivation as shown in Activity 02.

1. $\displaystyle t^3 y^{\,\prime\prime\prime}-6t^2 y^{\,\prime\prime}+19ty^{\,\prime}-27y=0$, $t>0$.