Chapter 2.2, Problem 39P

Variation of Parameters. Consider the following method of solving the general linear equation of first order:

$y\text{'}+p(t)y=g(t).$ (i)

If $g(t)=0$ for all $t$, show that the solution is

$y=A\exp \left[-{\displaystyle \int p\left(t\right)dt}\right]$ (ii)

Where $A$ is a constant.

If $g(t)$ is not everywhere zero, assume that the solution of Eq. (i) is of the form for all $t$, show that the solution is

$y=A(t)\exp \left[-{\displaystyle \int p\left(t\right)dt}\right]$ (iii)

Where $A$ is now a function of $t$. By substituting for y in the given differential equation, show that $A(t)$ must satisfy the condition

$A\text{'}(t)=g(t)\exp \left[{\displaystyle \int p\left(t\right)dt}\right]$ (iv)

Find $A(t)$ from Eq. (iv). Then substitute for $A(t)$ in Eq. (iii) and determine $y$. Verify that the solution obtained in this manner agrees with that of Eq. (28) in the text. This technique is known as the method of variation of parameters; it is discussed in detail in Section 4.7 in connection with second order linear equations.