NONHOMOGENEOUS EQUATIONS;
VARIATION of PARAMETERS METHOD

Consider equations of the form

$$L[y]=\frac{d^2y}{dt^2}+ p(t)\frac{dy}{dt}+ q(t)y = g(t),$$

given independent solutions y₁(t) and y₂(t) to L[y]=0.

General Solution:

use y(t) = c₁y₁(t)+c₂y₂(t)+Y(t),
where Y(t) is a particular solution to L[y]=g(t).

Variation of Parameters:

let

Y(t) = u₁(t)y₁(t)+u₂(t)y₂(t).

• Using some algebra it can be shown that
  
  u'₁(t)y₁(t)+u'₂(t)y₂(t) = 0,
  
  
  
  
  u'₁(t)y'₁(t)+u'₂(t)y'₂(t) = g(t).
  
  

• Then
  
  $$u'_1(t) = -\frac{y_2(t)g(t)}{W(y_1,y_2)(t)}\mbox{\ and\ } u'_2(t) = \frac{y_1(t)g(t)}{W(y_1,y_2)(t)},$$
  
  
  so
  
  $$\rule{0pt}{5pt}\hspace{-10mm} u_1(t)= -\displaystyle\int\fra... ... u_2(t)= \displaystyle\int\frac{y_1(s)g(s)}{W(y_1,y_2)(s)}ds.$$
  
  

• Final Y(t)
  
  $$\rule{0pt}{5pt}\hspace{-10mm} Y(t)= -y_1(t)\displaystyle\int... ... +\ y_2(t)\displaystyle\int\frac{y_1(s)g(s)}{W(y_1,y_2)(s)}ds,$$
  
  
  or
  
  $$Y(t)=\displaystyle\int\frac{y_1(s)y_2(t)-y_1(t)y_2(s)}{y_1(s)y'_2(s)-y'_1(s)y_2(s)}g(s)ds.$$