1902 The Exponential Response Formula Resonant Case

The starting point for understanding the mathematics of pure resonance is the generalized Exponential Response formula. First recall the simple case of the Exponential Response formula:
Asolution to $$p(D)x=Be^{at}\tag{1}$$ is given by $$x_p=\frac{Be^{at}}{p(a)}\ p(a)\neq0\tag{2}$$ In the sessionon Exponential Response we also saw the generalization of this formula when $p(a)=0$. Here we will need to use the special case when $p(a)\neq0$:
A solution to equation $(1)$ is given by $$x_p=\frac{Bte^{at}}{p'(a)}\text{ if } p(a)=0 \text{ and } p'(a)\neq0\tag{2}$$ We will call this the Resonant Response Formula.

Example. Find a particular solution to the DE $x''+4x=2\cos 2t$. As usual, we try complex replacement and the ERF:
if $z_p$ is a solution to the complex DE $z''+4z=2e^{2it}$, then $x_p$ will be a solution to $x''+4x=2\cos 2t$.
The characteristic polynomial is $p(s)=s^2+4$ and $a=2i$, so that we have $p(a)=0$.
But since $p'(s)=2s$, and we have $p'(a)=p'(2i)=4i\neq0$.
The resonant case of the ERF thus gives $$z_p=\frac{2te^{2it}}{4i}$$ Then taking the real part of $z_p$ gives us our particular solution $$\begin{aligned} x_p&=Re(z_p)\\ &=Re(\frac{2t}{4i}(\cos 2t + i\sin 2t))\\ &=\frac{t}{2}\sin 2t \end{aligned}$$