The Principle of Superposition

Suppose that we have a linear homogenous second order differential equation $\frac{d^2 y}{dt^2} + p(t) \frac{dy}{dt} + q(t)y = 0$ and that $y = y_1(t)$ and $y = y_2(t)$ are both solutions. The following theorem says that any linear combination $y = Cy_1(t) + Dy_2(t)$ is also a solution to this differential equation for any constants $C$ and $D$.

Theorem 1 (The Principle of Superposition for Second Order Differential Equations): If $\frac{d^2 y}{dt^2} + p(t) \frac{dy}{dt} + q(t) y = 0$ is a second order linear differentiation equation and $y = y_1(t)$ and $y = y_2(t)$ are both solutions to this differential equation, then for $C$ and $D$ as constants, $y = Cy_1(t) + Dy_2(t)$ is also a solution.

• Proof: Suppose that $y = y_1$ and $y = y_2$ are both solutions to the second order linear differential equation $\frac{d^2 y}{dt^2} + p(t) \frac{dy}{dt} + q(t) y = 0$. Then we have that:

(1)

\begin{align} \quad \frac{d^2 y_1}{dt^2} + p(t) \frac{d y_1}{dt} + q(t)y_1 = 0 \quad \mathrm{and} \quad \frac{d^2 y_2}{dt^2} + p(t) \frac{d y_2}{dt} + q(t) y_2 = 0 \end{align}

• If $C$ and $D$ are constants we we plug $y = Cy_1(t) + Dy_2(t)$ into our differential equation, we get that:

(2)

\begin{align} \quad \frac{d^2}{dt^2} \left ( Cy_1(t) + Dy_2(t) \right ) + p(t) \frac{d}{dt} \left ( Cy_1(t) + Dy_2(t) \right ) + q(t) \left ( Cy_1(t) + Dy_2(t) \right ) \\ = C \frac{d^2 y_1}{dt^2} + D \frac{d^2 y_2}{dt^2} + p(t) C \frac{d y_1}{dt} + p(t) D \frac{d y_2}{dt} + q(t) C y_1 + q(t) D y_2 \\ = C \underbrace{\left [ \frac{d^2 y_1}{dt^2} + p(t) \frac{d y_1}{dt} + q(t)y_1 \right ]}_{=0} + D \underbrace{\left [ \frac{d^2 y_2}{dt^2} + p(t) \frac{d y_2}{dt} + q(t)y_2 \right ]}_{=0} \\ = 0 \end{align}

• Therefore $y = Cy_1(t) + Dy_2(t)$ is also a solution to this differential equation. $\blacksquare$