Chapter 5.7, Problem 8P

(a)

To determine

To show: That the point $x=0$ a regular singular point of the given differential equation and roots of the indicial equation are $v$ and $-v$.

(b)

To determine

To show: That the first solutions is $y_1\left(x\right)=x^v\left[1-\frac{1}{1!\left(1+v\right)}\times {\left(\frac{x}{2}\right)}^2+\frac{1}{2!\left(1+v\right)\left(2+v\right)}\times {\left(\frac{x}{2}\right)}^4+{\displaystyle \sum _{m=3}^{\infty }\frac{{\left(-1\right)}^m}{m!\left(1+v\right)\cdot \cdot \cdot \left(m+v\right)}{\left(\frac{x}{2}\right)}^{2m}}\right]$.

(c)

To determine

To show: That the second solutions is $y_2\left(x\right)=x^{-v}\left[1-\frac{1}{1!\left(1-v\right)}\times {\left(\frac{x}{2}\right)}^2+\frac{1}{2!\left(1-v\right)\left(2-v\right)}\times {\left(\frac{x}{2}\right)}^4+{\displaystyle \sum _{m=3}^{\infty }\frac{{\left(-1\right)}^m}{m!\left(1-v\right)\cdot \cdot \cdot \left(m-v\right)}{\left(\frac{x}{2}\right)}^{2m}}\right]$.

(d)

To determine

To verify: That the power series in the expressions for $y_1\left(x\right)$ and $y_2\left(x\right)$ converge absolutely for all x.