Higher Order Homogenous Differential Eqs. - Constant Coefficients

Higher Order Homogenous Differential Equations - Constant Coefficients

Recall that if we have a second order linear homogenous differential equation with constant coefficients $a, b, c \in \mathbb{R}$, that is:

(1)

\begin{align} \quad a \frac{d^2y}{dt^2} + b \frac{dy}{dt} + cy = 0 \end{align}

We saw that the roots $r_1$ and $r_2$ of the characteristic equation $ar^2 + br + c = 0$ had importantance on the form of the solutions to our differential equation. In particular, if $r_1$ and $r_2$ were both real and distinct roots, then the solution to our differential equation was of the form $y = Ce^{r_1t} + De^{r_2t}$. If $r_1$ and $r_2$ were complex conjugate roots where $r_1 = \lambda + \mu i$ and $r_2 = \lambda - \mu i$, then the solution to our differential equation was of the form $y = Ce^{\lambda t}\cos (\mu t) + De^{\lambda t}\sin (\mu t)$. Lastly, if $r_1$ and $r_2$ were real nondistinct roots, that is $r_1 = r_2$ (the root of this characteristic equation has multiplicity two), then the solution to our differential equation was of the form $y = Ce^{r_1t} + Dte^{r_1t}$.

We will now begin to extend these ideas to higher order linear homogenous differential equations with constant coefficients.

Consider the following $n^{\mathrm{th}}$ order linear homogenous differential equation with constant coefficients $a_0, a_1, ..., a_n \in \mathbb{R}$:

(2)

\begin{align} \quad a_0 \frac{d^{n}y}{dt^{n}} + a_1 \frac{d^{n-1}y}{dt^{n-1}} + ... + a_{n-1} \frac{dy}{dt} + a_n y = 0 \end{align}

Like when we dealt with second order differential equations of this type, for specific values of $r$, we will have that $y = e^{rt}$ be a solution to our differential equation. Plugging this into the differential equation above and we have that:

(3)

\begin{align} \quad a_0 \frac{d^{n}}{dt^{n}}(e^{rt}) + a_1 \frac{d^{n-1}}{dt^{n-1}}(e^{rt}) + ... + a_{n-1} \frac{d}{dt}(e^{rt}) + a_n (e^{rt}) \\ \quad = a_0 r^ne^{rt}+ a_1r^{n-1}e^{rt}+ ... + a_{n-1}re^{rt} + a_n e^{rt} \\ \quad = e^{rt} \left ( a_0r^n + a_1r^{n-1} + ... + a_{n-1}r + a_n \right ) \end{align}

We want the equation above to be equal to zero so that $y = e^{rt}$ is a solution to our $n^{\mathrm{th}}$ order linear homogenous differential equation. This happens if and only if $e^{rt} = 0$ or $a_0r^n + a_1r^{n-1} + ... + a_{n-1}r + a_n = 0$. Note though that $e^{rt} \neq 0$ ever, and so more specifically, the roots of the polynomial $a_0r^n + a_1r^{n-1} + ... + a_{n-1}r + a_n$ will give us suitable values of $r$ for which $y = e^{rt}$ is a solution to our differential equation. This polynomial has an important name which we've already seen but redefine below.

Definition: The Characteristic Equation for the $n^{\mathrm{th}}$ order linear homogenous differential equation $a_0 \frac{d^{n}y}{dt^{n}} + a_1 \frac{d^{n-1}y}{dt^{n-1}} + ... + a_{n-1} \frac{dy}{dt} + a_n y = 0$ with constant coefficients $a_0, a_1, ..., a_n \in \mathbb{R}$ is $a_0r^n + a_1r^{n-1} + ... + a_{n-1}r + a_n$.

Some people prefer to use the term, "Auxiliary Equation" or "Characteristic Polynomial" to mean the same thing as "Characteristic Equation".

Note that if we have an $n^{\mathrm{th}}$ order linear homogenous differential equation with constant coefficients, then the coefficient $a_0 \neq 0$ (since otherwise we would have a lower order differential equation). Thus, the characteristic polynomial $a_0r^n + a_1r^{n-1} + ... + a_{n-1}r + a_n$ will be of degree $n$. By the Fundamental Theorem of Algebra, there will exist $n$ roots, not necessary all real and not necessarily all distinct to this characteristic polynomial, call them $r_1$, $r_2$, …, $r_n$. We will subsequently begin to look at the general solutions under various cases in which these roots vary.