Chapter 4.3, Problem 52P

To determine

To prove: The general solution of the Cauchy-Euler equation $a{x}^{2}y\text{'}\text{'}+bxy\text{'}+cy=0$ in $x>0$ is $y=c_1{e}^{{\lambda }_{1}t}+c_2{e}^{{\lambda }_{2}t}=c_1{x}^{{\lambda }_1}+c_2{x}^{{\lambda }_2}$, if ${\lambda }_1$ and ${\lambda }_2$ are real and ${\lambda }_1\ne {\lambda }_2$, $y=c_1{e}^{{\lambda }_{1}t}+c_{2}t{e}^{{\lambda }_{2}t}=c_1{x}^{{\lambda }_1}+c_2{x}^{{\lambda }_2}\ln x$, if ${\lambda }_1={\lambda }_2$ and $y=e^{\mu t}\left[c_1\cos vt+c_2\sin vt\right]=x^{\mu }\left[c_1\cos \left(v\ln x\right)+c_2\sin \left(v\ln x\right)\right]$, if ${\lambda }_1=\mu +iv$ and ${\lambda }_2=\mu -iv$.