91Ó°ÊÓ

Given that \(y=t^{-1} \sin 4 t\) is a solution of \(t y^{\prime \prime}+\) \(2 y^{\prime}+16 t y=0\), find and graph the solution of the equation that satisfies \(y(\pi / 8)=0\) and \(y^{\prime}(\pi / 8)=32\).

(Abel's Formula for Higher Order Equations) Consider the \(n\)th order linear homogeneous equation \(y^{(n)}+p_{n-1}(t) y^{(n-1)}+\cdots+\) \(p_{0}(t) y=0\) with \(n\) linearly independent solutions \(y_{1}, y_{2}, \ldots, y_{n}\) on an interval \(I\). We can show that the Wronskian of these \(n\) solutions satisfies the same identity as that presented in Exercises 4.1. We do this for the third order differential equation, \(y^{\prime \prime \prime}+\) \(p_{2}(t) y^{\prime \prime}+p_{1}(t) y^{\prime}+p_{0}(t) y=0\), with linearly independent solutions \(y_{1}, y_{2}\), and \(y_{3}\) on an interval \(I\). (a) Show that $$ \frac{d}{d t}\left(W\left(\left\\{y_{1}, y_{2}, y_{3}\right\\}\right)\right)=\left|\begin{array}{ccc} y_{1} & y_{2} & y_{3} \\ y_{1}^{\prime} & y_{2}^{\prime} & y_{3}^{\prime} \\ y_{1}^{\prime \prime \prime} & y_{2}^{\prime \prime \prime} & y_{3}^{\prime \prime \prime} \end{array}\right| . $$ (b) Use the differential equation to solve for \(y_{1}^{\prime \prime \prime}, y_{2}^{\prime \prime \prime}\), and \(y_{3}^{\prime \prime \prime}\). Substitute these values to obtain \(d W / d t+p_{2}(t) W=0\). (c) Solve this differential equation to find that \(W\left(\left\\{y_{1}, y_{2}, y_{3}\right\\}\right)=C e^{-\int p_{2}(t) d t}\). (d) Indicate how to generalize this result to the \(n\)th order linear homogeneous equation.

Consider the second order Cauchy-Euler equation \(x^{2} y^{\prime \prime}+B x y^{\prime}+y=0, x>0\), where \(B\) is a constant. (a) Find \(\lim _{x \rightarrow \infty} y(x)\) where \(y(x)\) is a general solution of the equation using the given restriction on \(B\). (b) Determine if the solution is bounded as \(x \rightarrow \infty\) in each case: (i) \(B=1\); (ii) \(B>1\); (iii) \(B<1\).