The Euler equation \(x^{2} y^{\prime \prime}+\) \(\alpha x y^{\prime}+\beta y=0\)
can be reduced to an equation with constant coefficients by a change of the
independent variable. Let \(x=e^{z},\) or \(z=\ln x,\) and consider only the
interval \(x>0 .\)
(a) Show that
$$
\frac{d y}{d x}=\frac{1}{x} \frac{d y}{d z} \quad \text { and } \quad
\frac{d^{2} y}{d x^{2}}=\frac{1}{x^{2}} \frac{d^{2} y}{d
z^{2}}-\frac{1}{x^{2}} \frac{d y}{d z}
$$
(b) Show that the Euler equation becomes
$$
\frac{d^{2} y}{d z^{2}}+(\alpha-1) \frac{d y}{d z}+\beta y=0
$$
Letting \(r_{1}\) and \(r_{2}\) denote the roots of \(r^{2}+(\alpha-1) r+\beta=0\),
show that
(c) If \(r_{1}\) and \(r_{2}\) are real and different, then
$$
y=c_{1} e^{r_{1} z}+c_{2} e^{r_{2} z}=c_{1} x^{r_{1}}+c_{2} x^{r_{2}}
$$
(d) If \(r_{1}\) and \(r_{2}\) are real and equal, then
$$
y=\left(c_{1}+c_{2} z\right) e^{r_{1} z}=\left(c_{1}+c_{2} \ln x\right)
x^{r_{1}}
$$
(e) If \(r_{1}\) and \(r_{2}\) are complex conjugates, \(r_{1}=\lambda+i \mu,\) then
$$
y=e^{\lambda z}\left[c_{1} \cos (\mu z)+c_{2} \sin (\mu
z)\right]=x^{\lambda}\left[c_{1} \cos (\mu \ln x)+c_{2} \sin (\mu \ln
x)\right]
$$
