Chapter 9.6, Problem 18E

In Problems 17 and 18, use the results of Problem 42 in Exercises 9.5 to find a general solution to the given Cauchy-Euler system for $t>0$.

$t{x}^{\prime }\left(t\right)=\left[\begin{array}{cc}-1& -1\\ 9& -1\end{array}\right]x\left(t\right)$

a. Show that the Cauchy-Euler equation $a{t}^2{y}^{″}+bt{y}^{\prime }+cy=0$ can be written as a Cauchy-Euler system.

$t{x}^{\prime }=Ax$ $\left(25\right)$

with a constant coefficient matrix A, by setting $x_1=y/t$ and

$x_2=y^{\prime }$.

b. Show that for $t>0$ any system of the form (25) with A an $n\times n$ constant matrix has nontrivial solutions of the form $x\left(t\right)=t^{r}u$ if and only if r is an eigenvalue of A and $u$ is a corresponding eigenvector.