Higher Order Homogenous Differential Equations

Like with first and second order differential equations, a nicer-to-work-with class of higher order differential equations are higher order homogenous differential equations which we define below.

Definition: An $n^{\mathrm{th}}$ order linear differential equation $\frac{d^ny}{dt^n} + p_1(t) \frac{d^{n-1}y}{dt^{n-1}} + ... + p_{n-1}(t) \frac{dy}{dt} + p_n(t)y = g(t)$ is said to be Homogenous if $g(t) = 0$, that is, $\frac{d^ny}{dt^n} + p_1(t) \frac{d^{n-1}y}{dt^{n-1}} + ... + p_{n-1}(t) \frac{dy}{dt} + p_n(t)y = 0$.

Once again, it should be noted that the definition of higher order linear homogenous differential equations are analogous to that of first and second order linear homogenous differential equations.

Now recall that if we have a second order linear homogenous differential equation $\frac{d^2y}{dt^2} + p(t)\frac{dy}{dt} + q(t)y = 0$, then if $y = y_1(t)$ and $y = y_2(t)$ are solutions to this differential equation, then any linear combination $y = Cy_1(t) + Dy_2(t)$ is also a solution to this differential equation. The following result extends this to higher order linear homogenous differential equations.

Theorem 1: If $\frac{d^ny}{dt^n} + p_1(t) \frac{d^{n-1}y}{dt^{n-1}} + ... + p_{n-1}(t) \frac{dy}{dt} + p_n(t)y = 0$ is an $n^{\mathrm{th}}$ order linear homogenous differential equation and $y = y_1(t)$, $y = y_2(t)$, …, and $y = y_n(t)$ are all solutions to this differential equation, then for $C_1$, $C_2$, …, $C_n$ as constants, $y = C_1y_1(t) + C_2y_2(t) + ... + C_ny_n(t)$ is also a solution.

It's not have to see that Theorem 1 holds for the case when $n = 1$. Furthermore, we have already proven that case when $n = 2$ on The Principle of Superposition page.