Repeated Roots of The Characteristic Equation

Higher Order Homogenous Differential Equations - Repeated Roots of The Characteristic Equation

Suppose that we have an $n^{\mathrm{th}}$ order linear homogenous differential equation with constant coefficients:

(1)
\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}

We have already seen forms of the general solutions for when the roots $r_1$, $r_2$, …, $r_n$ of the characteristic equation $a_0r^n + a_1r^{n-1} + ... + a_{n-1}r + a_n = 0$ are unique real and/or complex. The only case left is for finding the general solution for when some of real roots are repeated, and for when some of the complex roots are repeated.

First, suppose that we have a real root $r_1$ that is repeated $s$ times. Then this root is said to have multiplicity $s ≤ n$, and the corresponding linearly independent solutions for this root are $e^{r_1t}$, $te^{r_1t}$, … , $t^{s-2}e^{r_1t}$, and $t^{s-1}e^{r_1t}$. Thus if $p_1$, $p_2$, …, $p_l$ are repeated roots with respective multiplicities $s_1$, $s_2$, …, $s_l$, and if $m(t)$ is the remaining terms that aren't regarding repeated real roots, then the general solution to our differential equation is:

(2)
\begin{align} \quad y = \sum_{j=1}^{s_1} C_{j, 1} t^{j-1}e^{p_1t} + \sum_{j=1}^{s_2} C_{j, 2} t^{j-1}e^{p_2t} + ... + \sum_{j=1}^{s_l} C_{j, l} t^{j-1}e^{p_lt} + m(t)\\ \end{align}

Here we have the $C$'s are constants.

Similarly, suppose that we have a complex root $r_1$ that is repeated $s$ times. If $r_1 = \lambda + \mu i$, then we know that the corresponding complex conjugate, $\lambda - \mu i$ is also a complex root to the characteristic equation and it is also repeated $s$ times, and thus, we obtain $2s$ real-valued solutions corresponding to this pair of complex-conjugate repeated roots, $e^{\lambda t}\cos (\mu t)$, $e^{\lambda t} \sin (\mu t)$, $te^{\lambda t} \cos (\mu t)$, $te^{\lambda t} \sin (\mu t)$, …, $t^{s-1} e^{\lambda t} \cos (\mu t)$, $t^{s-1} e^{\lambda t}\sin (\mu t)$.

Thus if $q_1$, $q_2$, …, $q_m$ are repeated complex roots, corresponding to $\lambda_r \pm \mu_r$ for $r = 1, 2, ... s$, and each having multiplicity $s_1$, $s_2$, …, $s_m$ respectively, and if $m(t)$ is the remaining terms that aren't regarding the repeated complex roots, then the general solution to our differential equation is:

(3)
\begin{align} \quad y = \sum_{j=1}^{s_1} t^{j-1} e^{\lambda_1t} \left ( C_{j, 1} \cos (\mu_1 t) + D_{j,1} \sin (\mu_1 t) \right ) + \sum_{j=1}^{s_2} t^{j-1} e^{\lambda_2t} \left ( C_{j, 2} \cos (\mu_2 t) + D_{j,2} \sin (\mu_2 t) \right ) + ... \\ + \sum_{j=1}^{s_m} t^{j-1} e^{\lambda_mt} \left ( C_{j, m} \cos (\mu_m t) + D_{j,m} \sin (\mu_m t) \right ) + m(t) \end{align}

Here we have that the $C$'s and $D$'s are constants