HIGHER ORDER EQUATIONS:
VARIATION of PARAMETERS

General Problem Form:

n^th order equation

$$L[y] = \frac{d^ny}{dt^n} + p_1(t)\frac{d^{n-1}y}{dt^{n-1}} + \cdots + p_n(t)y = g(t),$$

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).

Variation of Parameters

: assume

$$Y(t) = u_1(t)y_1(t)+u_2(t)y_2(t)+\cdots+u_n(t)y_n(t) .$$

• Solve the system $Y{\bf u}'={\bf g}$:
  
  $$\left[\begin{array}{llll} y_1&y_2&\ldots&y_n\\ y'_1&y'_2&\ldo... ...\left[\begin{array}{l}0\\ \vdots\\ 0\\ g(t)\end{array}\right].$$
  
  
  for $u'_1, u'_2, \ldots, u'_n$.
• Integrate $u'_1, u'_2, \ldots, u'_n$ to find $u_1, u_2, \ldots, u_n$.
• Solution exists if the Wronskian $W = det(Y) \neq 0$.
• $W(t) = Ce^{-\displaystyle\int p_1(t)}$.
• Let W_i be the determinant of the matrix obtained by replacing column i of Y by $[0, \ldots, 0, 1]^T$. Then
  
  $$u_i(t) = \displaystyle\int_{t_0}^t\frac{g(s)W_i(s)}{W(s)}ds,$$
  
  
  for $i = 1, 2, \ldots, n$.