# Reduction of Order on Second Order Linear Homogenous Differential Equations

Recall from the Repeated Roots of The Characteristic Equation page that if we had a second order linear homogenous differential equations with constant coefficients, (that is a differential equation in the form $a \frac{d^2 y}{dt^2} + b \frac{dy}{dt} + cy = 0$) where $a, b, c \in \mathbb{R}^2$, and if the roots $r_1, r_2$ of the characteristic equation $ar^2 + br + c = 0$ where real repeated roots, then a fundamental set of solutions could be constructed as:

(1)

Recall that finding one solution, namely $y_1(t) = e^{r_1t}$ can relatively easy. In finding a second solution $y = y_2(t)$ to form a fundamental set of solutions, we assume that $y = v(t) y_1(t)$ was a solution to this differential equation and then solved for $v(t)$. This technique can be applied to more general second order linear homogenous differential equations that will allow us to, in a sense, “convert” a second order linear differential equation to a first order linear differential equation which is often much more manageable to solve. This technique is known as **Reduction of Order** for differential equations.

Consider the second order linear homogenous differential equation $\frac{d^2y}{dt^2} + p(t) \frac{dy}{dt} + q(t) y = 0$, and suppose that $y = y_1(t)$ is a nonzero solution to this differential equation, and assume that $y = v(t) y_1(t)$ is also a solution to this differential equation. The first and second derivatives of $y = v(t) y_1(t)$ are:

(2)

(3)

If we plug $y = v(t) y_1(t)$, $y’ = v(t)y_1′(t) + v'(t) y_1(t)$ and $y” = v(t)y_1”(t) + 2v'(t)y_1′(t) + v”(t)y_1(t)$ into our differential equation, then we have that:

(4)

The differential equation above is rather nice because we can solve it as though it were a first order differential equation for the function $v’$.

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