Infinite Series Solution for Differential Equations

Infinite Series Solutions is the One of the most helpful technique often used in solving differential Equations. They are called special functions and the method is known as "Frobenius method". It is an improvisation of the common "power series method". 

    Some of the famous solutions are known as Legendre function, Bessel function, Hermite functions and etc. These are just the name given for the solutions of different differential equations where else the method is quite general.  

   Let us say, for example, we encounter a differential equation like, $$ {d^2}y/d{x^2} - 2x dy/dx + 2ny = 0 $$ which is called the Hermite differential equation. You may need to solve this equation, if you try to solve Schrodinger equation for Simple Harmonic Oscillator. 

   To proceed,  first we are going to assume, there exists really a solution, for this differential equation which can be expressed in terms of power series solution. 

   Then, we will insert that solution in our differential equation and impose the necessary conditions to be the solution and finally will get our modified output solution. 
Note: Remember, here we just seek a solution that fulfills our conditions, not anything more.   

   The general input solution is given by, $$ y(x) = \sum_{m=0}^{\infty} C_m x^{m+r} ........ where C_0 \neq 0 $$ Substituting this on our differential equation, our differential equation gets modified into $$ \sum_{m=0}^\infty \left[ (m+r)(m+r-1) C_m x^{m+r-2} + 2 [ n - (m+r)] C_m x^{m+r} \right] = 0 $$ This should be equal to zer0, which can be obtained only if and only if all the coefficients are zero. 

   Equating to zero, the coefficients of,
$$ (1) x^{r-2}, \ldots... r(r-1) = 0 \rightarrow r=0 \; or \; r=1 \; since \; C_0 \neq 0 $$ $$ (2) x^{r-1}, \ldots... (r+1)rC_1 = 0 .... \; if \;\; r=0, \; C_1\neq 0 ; and \; if \;\; r=1, \; then \; C_1=0 $$ $$ (3) x^{m+r}, \\~\\ for \; r=0 \dots......... \frac{C_{m+2}}{C_m} = \frac{-2(n-m)} {(m+1)(m+2)} \\~\\ for \; r=1 \dots...........\frac {C_{m+2}}{C_m} = \frac{-2[n-m-1]}{(m+2)(m+3)}$$

  Considering r=0, the general expression for even coefficients expressed as, 
$$ C_{2s} = \frac{(-1)^s 2^s n (n-2).....(n-2s+2)}{(2s!)} C_0 $$
and the odd coefficients are expressed as,
$$ C_{2s+1} = \frac{ (-1)^s 2^s (n-1) (n-3).....(n-2s+1)}{(2s+1)!} C_1$$
The general solution is therefore,

$$ y(x) = C_0 \left[ 1 + \sum_{s=1}^\infty \frac{ (-1)^s 2^s n (n-2)....(n-2s+2)}{(2s!)} x^{2s}\right] + \\~\\ C_1x \left[ 1 + \sum_{s=1}^\infty \frac{(-1)^s 2^s (n-1)(n-3)....(n-2s+1)}{(2s+1)!} x^{2s+1} \right] $$   

For r=1, if we proceed like the same, we will get a solution exactly similar to the second series in the general solution. Since its property already inherent in the general solution, we don't need to put much attention on that. 

   That's it. We arrived to our general solution. Depending on the   

nature of the problem, we can make the series to converge or stop by appropriately choosing the values of $ \; C_0 \;and \;C_1$ 

   There is a conventional way of choosing the constants, which gives the series of many function known as Hermite polynomials. It should be dealt separately on  how the Hermite polynomials are derived from the general solution. 

   As far as now, the general solution is just an example of how to get an output from a differential equation. To understand this more, we need to separately solve more differential equations in different physical situations.