# 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

### Definition: Variable-Coefficient Equation

$\displaystyle y^{\,\prime\prime}(t) + p(t)y^{\,\prime}(t) +q(t)y(t)=g(t)$.

### Theorem: Existence and Uniqueness of Solutions

Suppose $p(t)$, $q(t)$, and $g(t)$ are continuous on $(a,b)$ that contains $y_0$. Then, for any choice of the initial values of $Y_0$ and $Y_1$, there exists a unique solution in the interval to the IVP $\displaystyle y^{\,\prime\prime}(t)+p(t)y^{\,\prime}(t)+q(t)y(t)=g(t)$, $\displaystyle y(t_0)=Y_0$, $\displaystyle y^{\,\prime}(t_0)=Y_1$.

### Definition: Cauchy-Euler Equations, a.k.a., Equidimensional Equations

For real numbers $a$, $b$, and $c$, $\displaystyle at^2y^{\,\prime\prime}(t)+bty^{\,\prime}(t)+cy(t)=f(t)$. Cauchy-Euler Equations are equidimensional because $at^2y^{\,\prime\prime}(t)$, $bty^{\,\prime}(t)$, and $cy(t)$ have the dimensions (i.e., units).

#### Investigation 01

Is $y(t)=e^{rt}$ a solution to $\displaystyle at^2y^{\,\prime\prime}(t)+bty^{\,\prime}(t)+cy(t)=0$? Explain.