Higher-order Homogeneous Linear ODE with Constant Coefficients

Definition

$L(y)=y^{(n)}+a_{n-1}{y}^{(n-1)}+\cdots +a_1{y}^{\prime }+a_{0}y=0$ on $I$

Solution

Try $y=e^{\lambda x}\Rightarrow L(e^{\lambda x})=({\lambda }^n+a_{n-1}{\lambda }^{n-1}+\cdots ++a_1\lambda +a_0)$^[characteristic equation] $\underset{\ne 0}{\underbrace{e^{\lambda x}}}=0$

If $\lambda ={\lambda }_i$ for some $i=1,\dots ,n$ $\Rightarrow L(e^{{\lambda }_x})=0\Rightarrow e^{{\lambda }_{ix}}$ is a solution for $i=1,\dots ,n$

simple distinct real roots

${\lambda }_1,\dots ,{\lambda }_m\Rightarrow y=e^{{\lambda }_{1}x},\dots ,e^{{\lambda }_{m}x}$

simple complex roots

$\lambda =\gamma \pm i\omega \Rightarrow y=e^{\gamma x}\cos (\omega x),e^{\gamma x}\sin (\omega x)$

repeated real root

$\lambda \Rightarrow y=e^{\lambda x},x{e}^{\lambda x},\dots ,x^{s-1}{e}^{\lambda x}$

repeated complex root

$\begin{array}{rl}\lambda =\gamma \pm i\omega \Rightarrow y=& e^{\gamma x}\cos (\omega x),x{e}^{\gamma x}\cos (\omega x),\dots ,x^{s-1}{e}^{\gamma x}\cos (\omega x),\\ & e^{\gamma x}\sin (\omega x),x{e}^{\gamma x}\sin (\omega x),\dots ,x^{s-1}{e}^{\gamma x}\sin (\omega x)\end{array}$