3.6 Variation of Parameters

In this section，we will intoduce the gerneral method in principle at least, it can be applied

to any equation, and it requires no detailed assumptions about the form of the solution.

Method of Variation of Parameters

In gerneral, we consider

$$y'' + p(t)y' + q(t)y = g(t)$$

where $p$ , $q$ and $g$ are continuous functions. We assume that we know

$$y_h = c_1y_1(t) + c_2 y_2(t)$$

In this idea, we replace the constant $c_1$,$c_2$ by $u_1(t)$,$u_2(t)$ $\text{resp.}$

$$y = u_1(t) y_1(t) + u_2(t) y_2(t) \quad --(a)$$ $$y' = u_1'(t) y_1(t) + u_1(t)y_1'(t) + u'_2(t) y_2(t) + u_2(t)y_2'(t)$$

Assume that $u_1'(t) y_1(t) + u'_2(t) y_2(t) = 0$, we have

$$y' = u_1(t)y_1'(t) + u_2(t)y_2'(t)$$ $$y'' = u_1'(t)y_1'(t) + u_1(t) y_1''(t) + u_2'(t)y_2'(t) + u_2(t)y''_2(t)$$

Now, we substitute$y$,$y'$and $y''$to the orginal equation. We find that

$$\begin{aligned} u_1(t(y_1''(t)+p(t)y_1'(t)+q(t)y_1(t)) \\ +u_2(t(y_2''(t)+p(t)y_2'(t)+q(t)y_2(t)) \\ + u_1'(t)y_1'(t) + u_2'(t)y_2'(t) = g(t)\\ \end{aligned}$$

where $y_1$and $y_2$are the homogeneous solutions ,hence the equation above can be reduced to

$$u_1'(t)y_1'(t) +u_2'(t)y_2'(t) = g(t) \quad --(b)$$

Since (a) and (b) forms a system of two linear algebraic equations for derivatives $u_1'(t)$ and $u_2'(t)$of the unknown function. By Carmer's rule, we have:

$$u_1'(t) = -\frac{y_2(t)g(t)}{W[y_1 ,y_2](t)} ,\quad u_2'(t) = -\frac{y_1(t)g(t)}{W[y_1 ,y_2](t)}$$

Integrate both of them, and substitude into (a) ,you can get the answer.