MIT 1803 lecture 13

This lecture covered finding the $y_p$, a particular solution of the ODE

$y'' + Ay' + By = f(x)$

Let the solutions be generally complex so that $e^{(a+iw)x} = e^{\alpha x}$.   Express the ODE as

$D^2y + A D y + B y = f(x)$

let $p(D)$ be $(D^2 + AD + B)$ or any polynomial operator on $D$, then

$p(D) e^{\alpha x} = p(\alpha) e^{\alpha x}$

which we will call the substitution rule.  This yields the exponential input theorem

$y_p = \frac{e^{\alpha x}}{p(\alpha) }$

when $p(\alpha) \neq 0$.   If $p(\alpha) = 0$, then the exponential shift rule can be used

$p(D) e^{\alpha x} u(x) = e^{\alpha x} p(D + \alpha) u(x)$

In proving this rule – he again warns you not to hack your way forward – generating more work than necessary – make sure to use your previous result.