Exercise

1. Solve the following differential equation by the reduction of order.
$$\begin{array}{ll} (a) \ y^{\prime\prime} - 3y^{\prime} + 2y =... ...\ y^{\prime\prime} + y = \sec{x}, \ y_{1} = \cos{x} \end{array}$$
2. Suppose $y_{1}(x)$ is a solution of $y^{\prime\prime} + a_{1}(x)y^{\prime} + a_{0}(x)y = 0$. Then an another solution $y_{2}(x)$ is given by

$$y_{2}(x) = y_{1}(x)\int \frac{e^{-\int a_{1}(x)dx}}{y_{1}^{2}} dx.$$

Also, show that $\{y_{1}$ and $y_{2}\}$ are linearly independent.