4.2: The Principle of Superposition

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Consider the linear second-order homogeneous ode:

\[\label{eq:1}\overset{..}{x}+p(t)\overset{.}{x}+q(t)x=0;\] and suppose that \(x = X_1(t)\) and \(x = X_2(t)\) are solutions to \(\eqref{eq:1}\). We consider a linear combination of \(X_1\) and \(X_2\) by letting \[\label{eq:2}X(t)=c_1X_1(t)+c_2X_2(t),\] with \(c_1\) and \(c_2\) constants. The principle of superposition states that \(x = X(t)\) is also a solution of \(\eqref{eq:1}\). To prove this, we compute \[\begin{aligned} \overset{..}{X}+p\overset{.}{X}+qX&=c_1\overset{..}{X}_1+c_2\overset{..}{X}_2+p(c_1\overset{.}{X}_1+c_2\overset{.}{X}_2)+q(c_1X_1+c_2X_2) \\ &=c_1(\overset{..}{X}_1+p\overset{.}{X}_1+qX_1)+c_2(\overset{..}{X}_2+p\overset{.}{X}_2+qX_2) \\ &=c_1\times 0+c_2\times 0 \\ &=0,\end{aligned}\] since \(X_1\) and \(X_2\) were assumed to be solutions of \(\eqref{eq:1}\). We have therefore shown that any linear combination of solutions to the homogeneous linear second-order ode is also a solution.

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