8.2: The Principle of Superposition
Consider the homogeneous linear second-order ode:
\[\ddot{x}+p(t) \dot{x}+q(t) x=0 \nonumber \]
and suppose that \(x=X_{1}(t)\) and \(x=X_{2}(t)\) are solutions to Equation \ref{8.4}. We consider a linear combination of \(X_{1}\) and \(X_{2}\) by letting
\[X(t)=c_{1} X_{1}(t)+c_{2} X_{2}(t), \nonumber \]
with \(c_{1}\) and \(c_{2}\) constants. The principle of superposition states that \(x=X(t)\) is also a solution of Equation \ref{8.4}. To prove this, we compute
\[\begin{aligned} \ddot{X}+p \dot{X}+q X &=c_{1} \ddot{X}_{1}+c_{2} \ddot{X}_{2}+p\left(c_{1} \dot{X}_{1}+c_{2} \dot{X}_{2}\right)+q\left(c_{1} X_{1}+c_{2} X_{2}\right) \\ &=c_{1}\left(\ddot{X}_{1}+p \dot{X}_{1}+q X_{1}\right)+c_{2}\left(\ddot{X}_{2}+p \dot{X}_{2}+q X_{2}\right) \\ &=c_{1} \times 0+c_{2} \times 0 \\ &=0 \end{aligned} \nonumber \]
since \(X_{1}\) and \(X_{2}\) were assumed to be solutions of Equation \ref{8.4}. We have therefore shown that any linear combination of solutions to the homogeneous linear second-order ode is also a solution.