Chapter 5, Problem 27CR

To determine

To show: The general solution of the equation $y^{″}-2x{y}^{\prime }+2\alpha y=0$ is $y\left(x\right)=c_0{y}_1\left(x\right)+c_1{y}_2\left(x\right)$, where $y_1\left(x\right)=1+{\displaystyle \sum _{k=1}^{\infty }{\left(-1\right)}^k\frac{2^k\alpha \left(\alpha -2\right)\cdots \left(\alpha -2k+2\right)}{\left(2k\right)!}}{x}^{2k}$ and $y_2\left(x\right)=x+{\displaystyle \sum _{k=1}^{\infty }{\left(-1\right)}^k\frac{2^k\left(\alpha -1\right)\left(\alpha -3\right)\cdots \left(\alpha -2k+1\right)}{\left(2k+1\right)!}}{x}^{2k+1}$ are power series solutions centered at the ordinary point $0$.