Reduction of Order on Second Order Linear Homogenous Diff. Eqs.

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)
\begin{align} \quad y = Ce^{r_1t} + Dte^{r_1t} \end{align}

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)
\begin{align} \quad y' = v(t)y_1'(t) + v'(t) y_1(t) \end{align}
(3)
\begin{align} \quad y'' = v(t)y_1''(t) + v'(t)y_1'(t) + v'(t)y_1'(t) + v''(t) y_1(t) \\ \quad y'' = v(t)y_1''(t) + 2v'(t)y_1'(t) + v''(t)y_1(t) \end{align}

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)
\begin{align} \quad \left [ v(t)y_1''(t) + 2v'(t)y_1'(t) + v''(t)y_1(t) \right ] + p(t) \left [ v(t)y_1'(t) + v'(t) y_1(t) \right ] + q(t) \left [ v(t) y_1(t) \right ] = 0 \\ \quad \underbrace{\left ( y_1''(t) + p(t)y_1'(t) + q(t) y_1(t) \right )}_{=0} v(t) + \left ( 2y_1'(t) + p(t) y_1(t) \right ) v'(t) + y_1(t) v''(t) = 0 \\ \quad \left ( 2y_1'(t) + p(t) y_1(t) \right ) v'(t) + y_1(t) v''(t) = 0 \end{align}

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