Variation of Parameters

For

$$a{y}^{\prime \prime }+b{y}^{\prime }+cy=f(t)$$

1. Find two solutions ($y_1$ and $y_2$) to the Homogeneous Linear Equation
2. Compute the Wronskian $W$, $W_1$, and $W_2$ using Cramer’s Rule
3. $v_1=\int \frac{W_1}{W}dt$ and $v_2=\int \frac{W_2}{W}dt$
   • We do not need constants because we are just finding an anti-derivative
4. $y_p=v_1{y}_1+v_2{y}_2$

$$\begin{array}{rl}W& =\left|\begin{array}{cc}{y}_1& y_2\\ y_1^{\prime }& y_2^{\prime }\end{array}\right|\\ W_1& =\left|\begin{array}{cc}0& y_2\\ \frac{f(t)}{a}& y_2^{\prime }\end{array}\right|\\ W_2& =\left|\begin{array}{cc}{y}_1& 0\\ y_1^{\prime }& \frac{f(t)}{a}\end{array}\right|\end{array}$$

If $f(t)$ a linear combination of functions, you can do variation of parameters for each of the functions and then the partial solution just has all of them because of The Superposition Principle