Chapter 6.2, Problem 31E

Higher-Order Cauchy-Euler Equations. A differential equation that can be expressed in the form

$a_n{x}^n{y}^{\left(n\right)}\left(x\right)+a_{n-1}{x}^{n-1}{y}^{\left(n-1\right)}\left(x\right)+\cdots +a_{0}y\left(x\right)=0$,

where $a_n,a_{n-1},\dots ,a_0$ are constants, is called a homogeneous Cauchy-Euler equation. (The second-order case is discussed in Section 4.7.) Use the substitution $y=x^r$ to help determine a fundamental solution set for the following Cauchy-Euler equations:

a. $x^3{y}^{‴}+x^2{y}^{″}-2x{y}^{\prime }+2y=0$, $x>0$.

b. $x^4{y}^{\left(4\right)}+6x^3{y}^{‴}+2x^2{y}^{″}-4x{y}^{\prime }+4y=0$, $x>0$

c. $x^3{y}^{‴}-2x^2{y}^{″}+13x{y}^{\prime }-13y=0$, $x>0$

[Hint: $\begin{array}{rll}{x}^{\alpha +i\beta }& =& e^{\left(\alpha +i\beta \right)\ln x}\\ & =& x^{\alpha }\left\{\cos \left(\beta \ln x\right)+i\sin \left(\beta \ln x\right)\right\}\end{array}$.]