HIGHER ORDER EQUATIONS:
UNDETERMINED COEFFICIENTS

General Problem Form:

n^th order equation

$$L[y] = a_0\frac{d^ny}{dt^n} + a_1\frac{d^{n-1}y}{dt^{n-1}} + \cdots + a_ny = g(t),$$

given independent solutions $y_1(t), \ldots, y_n(t)$ to L[y]=0.

General Solution

is

$$y(t) = c_1y_1(t)+c_2y_2(t)+\cdots+c_ny_n(t)+Y(t),$$

where Y(t) is a particular solution to L[y]=g(t).

Superposition of Solutions:

if $g = g_1 + \cdots + g_n$, and
L[Y_i] = g_i, for $i=1,\ldots ,n$, then $Y = Y_1 + \cdots + Y_n$.

Undetermined Coefficients

• Consider each g_i separately and sum results to get Y.
• Standard cases for g_i(t): let $P_n(t) = p_0t^n + \cdots + p_n$.

  gi(t)	Yi(t)
  Pn(t)	tsQn(t)
  $P_n(t)e^{\alpha t}$	$t^sQ_n(t)e^{\alpha t}$
  $P_n(t)e^{\alpha t}(\sin(\beta t)\mbox{\ or\ }\cos(\beta t))$	$t^se^{\alpha t}(Q_n(t)\sin(\beta t)+R_n(t)\cos(\beta t))$

  s is the smallest nonnegative integer for which every term in Y_i(t)is different from all of the y_i(t)'s.