Question

1. Consider y′′ + ω2y = A cos ωx. A and ω are positive constants. (a) Find all solutions 0 ≤ x ≤ ∞.
(b) Show that every for solution, |yi(x)| assumes arbitrarily large values as x → ∞.

Answer 1

We start by finding a solution for the homogeneous equation\[y''+\omega^2y = 0.\]The characteristic equation is\[\lambda^2+\omega^2=0\Rightarrow\lambda = \pm i\omega\]so \[y_h(x) = C_1 \sin(\omega x) + C_2 \cos(\omega x)\]is the general solution for this equation. Now we seek a particular solution of the original ODE. The Wronskian is\[W(x) = \left| \begin{array}&\cos (\omega x)&\sin (\omega x)\\-\omega\sin (\omega x)&\omega\cos (\omega x)\end{array} \right| = \omega\]and\begin{eqnarray*}y_p(x) &=& -\cos (\omega x) \int \frac{A \cos{(\omega x)} \sin (\omega x)}{\omega}dx+\sin (\omega x) \int \frac{A \cos{(\omega x)} \cos (\omega x)}{\omega}dx \\ &=& \frac{A}{2\omega^2}(\cos(\omega x) + \omega x \sin{(\omega x)})\end{eqnarray*}finally, the general solution is\[y(x) = C_1 \cos(\omega x) + C_2 \sin(\omega x) + A \omega x \sin(\omega x).\]It's evident that this function assumes arbitrarily large values for large x.