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{\Large\bf The Variation of Parameters Formula}

\vspace{.3in}

Variation of Parameters is a method for computing a particular
solution to the nonhomogeneous linear second-order ode:
$$
y''(t)+p(t)y'(t)+q(t)y(t)=f(t)
$$

\vspace{.3in}

{\bf Solution Procedure}

\vspace{.2in}

There are two steps in the solution procedure: 

\begin{itemize}

\item[(1)] Find two linearly independent solutions to the homogeneous
problem

$$
y''(t)+p(t)y'(t)+q(t)y(t)=0
$$

Call these solutions y1(t) and y2(t). 

\item[(2)] Use the variation of parameters formula to determine
the particular solution: 

$$
y_p(t)=-y_1(t)\int\frac{y_2(t)f(t)}{W(t)}dt+
y_2(t)\int\frac{y_1(t)f(t)}{W(t)}dt
$$

where W(t), called the Wronskian, is defined by

$$
W(t)=y_1(t)y'_2(t)-y_1'(t)y_2(t)
$$

According to the theory of second-order ode, the Wronskian is
guaranteed to be non-zero, if y1(t) and y2(t) are 
linearly independent.

\end{itemize}


\vspace{.3in}

{\bf Example}

\vspace{.2in}

Consider, for example, the ode  

$$
y''-3y'-18y=t
$$

The homogeneous equation is
$$
y''-3y'-18y=0
$$
Two linearly-independent solutions to the ode are:

$$
y_1(t)=e^{6t} \hspace{.2in} y_2(t)=e^{-3t}
$$

The Wronskian is:

$$
W(t)=y_1y_2'-y_1'y_2=e^{6t}(-3e^{-3t})-(6e^{6t})e^{-3t}=-9e^{3t}
$$

The particular solution is

$$
y_p=-e^{6t}\int\frac{e^{-3t}t}{-9e^{3t}}dt+
e^{-3t}\int\frac{e^{6t}t}{-9e^{3t}}dt
$$

The integrals can be determined using integration by parts.
The first integral is

$$
-\frac{1}{9}\int te^{-6t}dt=\frac{1}{54}te^{-6t}+\frac{1}{324}e^{-6t}
$$

The second integral is

$$
-\frac{1}{9}\int te^{3t}dt=-\frac{1}{27}te^{3t}+\frac{1}{81}e^{3t}
$$

Putting everything together, we have:

$$
y_p=-e^{6t}(\frac{1}{54}te^{-6t}+\frac{1}{324}e^{-6t}) 
e^{-3t}(-\frac{1}{27}te^{3t}+\frac{1}{81}e^{3t}) 
=-\frac{1}{18}t+\frac{1}{108}
$$

The general solution to the differential equation is

$$
y(t)=C_1e^{6t}+C_2e^{-3t}-\frac{1}{18}t+\frac{1}{108}
$$

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