(Solved): Advanced ordinary differential equations Project 10: The proof of the Frobenius Theorem and Bes ...

Advanced ordinary differential equations

Project 10: The proof of the Frobenius Theorem and Bessel's functions a) Prove the following theorem of Frobenius solution at a regular singular point. Let $x=0$ be a regular singular point of the ODE $\frac{d^{2}y}{d{x}^2}+p(x)\frac{dy}{dx}+q(x)y=0$ with the analytic (at 0 ) functions $xp(x)={?}_{n=0}^{?}{p}_n{x}^n\text{ and }{x}^{2}q(x)={?}_{n=0}^{?}{q}_n{x}^n$ having radii of convergence $R_p$ and $R_q$ respectively. Let $r_1$ and $r_2$ be the roots of the Indicial equation, $r^2+\left(p_0?1\right)r+q_0=0$ If $r_1$ and $r_2$ are real and $r_1?r_2$ then, $y_1(x)=x^{r_1}{?}_{n=0}^{?}{a}_n{x}^n,a_0?=0$ Is a solution of the ODE (3). The form of the second linearly independent solution depends on $r_2$ as follows: 1. $r_1?r_2$ is not an integer $y_2(x)=x^{r_2}{?}_{n=0}^{?}{b}_n{x}^n,b_0?=0$ 2. $r_1=r_2=r$ $y_2(x)=y_1(x)\ln x+x^r{?}^{?}{c}_n{x}^n$ 3. $r_1?r_2$ is an integer In this case, the smaller root $r_2$ leads to both solutions or neither If neither; $y_2(x)=k{y}_1(x)\ln x+x^{r_2}{?}_{n=0}^{?}{c}_n{x}^n$ If $r_1$ and $r_2$ are complex conjugates, then solutions are similar to case 1 . b) Apply the Frobenius theorem to find series solutions centered at $x=0$ of the Bessel's equation of order $v$ given below and define the Bessel functions. Also discuss the general solution to the Bessel's equation. $x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+\left(x^2?v^2\right)y=0$