QUESTION $5.$ This problem has applications in cryptography, the process of hiding information

using codes so that only those for whom the message was intended can read or access it$.$ Cryptography

is an important field of study in today's digital age because it safeguards digital communication and

information from the constant threat of cyber$-$attacks and data breaches. Robust cryptographic meth$-$

ods continue to be developed to encrypt sensitive information to secure online transactions, protect

personal data, and maintain trust in digital systems.

Consider the second$-$order linear ODE

$(1+ax)y^{\text{'}\text{'}}(x)-2x{y}^{\text{'}}(x)+(-1)y(x)=0$

where a and $$ are real constants.

Consider the case $a=\sqrt[\phantom{2}]{}.$ For which values of $x$ can you expect a unique and twice differentiable

solution to the IVP where

$y(0)=0,y^{\text{'}}(0)=1$

Consider the case $a=0$ and $=2n+1,n=0,1,2,3,..$ is an odd integer. This case has

important applications in cryptography $($e$.$g$.,$ Salih $(2022)$ Optimizing RSA cryptosystem using

Hermite polynomials. Int. J$.$ Nonlinear Anal. Appl. $13$; Vinaroz et al $(2022)$ in Figure $2)$

$($a$)$ Find two linearly independent power series solutions for $y(x)$ about $x=0.$ Express one

series solution, $y_1(x),$ as one with even powers; while the other, $y_2(x),$ has odd powers.

$($b$)$ Use the recurrence relation to find the interval of convergence for each solution.

$($c$)$ Show that if $n$ is a positive integer, one of the series solutions is a polynomial? Write down

the polynomial for the case $n=6.$