Repeated Roots of The Characteristic Equation Examples 1

Recall from the Repeated Roots of The Characteristic Equation page that for a first order linear homogenous differential equation with constant coefficients, $a \frac{d^2 y}{dt^2} + b \frac{dy}{dt} + cy = 0$ where $a, b, c \in \mathbb{R}$, then if the roots $r_1$ and $r_2$ of the characteristic equation $ar^2 + br + c = 0$ are real and not distinct, that is $r_1 = r_2$, or simply just denote this root as $r$, then the general solution of this differential equation is given by:

(1)

\begin{align} \quad y = Ce^{rt} + Dte^{rt} \end{align}

We will now look at some examples of solving differential equations of this type.

Example 1

Find the general solution of the differential equation $\frac{d^2y}{dt^2} + 4 \frac{dy}{dt} + 4 y = 0$. Describe the behaviour of the general solution as $t$ approaches infinity.

We have that the characteristic equation for this differential equation is given by $r^2 + 4r + 4 = 0$. We can easily factor the characteristic equation as $(r + 2)(r + 2) = (r + 2)^2 = 0$. Therefore there is only one distinct root to our characteristic equation, that being $r = 2$. Therefore the general solution to this differential equation is:

(2)

\begin{align} \quad y = Ce^{2t} + Dte^{2t} \end{align}

As $t \to \infty$ we have that $e^{2t} \to \infty$ and so it should be relatively clear that $\lim_{t \to \infty} \left ( Ce^{2t} + Dte^{2t} \right ) = \infty$. The following image is a graph of a few of the functions to this differential equation.

Example 2

Find the general solution of the differential equation $\frac{d^2y}{dt^2} + 6 \frac{dy}{dt} + 9 y = 0$. Describe the behaviour of the general solution as $t$ approaches infinity.

The characteristic equation for this differential equation is $r^2 +6r + 9 = 0$. This characteristic equation can be easily factored as $(r + 3)(r + 3) = (r + 3)^2 = 0$. Therefore the repeated root in the characteristic equation is $r = -3$ and the general solution to our differential equation is:

(3)

\begin{align} \quad y = Ce^{-3t} + Dte^{-3t} \end{align}

Note that as $t \to \infty$, $e^{-3t} \to 0$. Exponential functions take "control" over linear functions, and so $\lim_{t \to \infty} \left ( Ce^{-3t} + Dte^{-3t} \right )= 0$. The following is a graph of a few solutions to this differential equation.