How do you solve this problem? More importantly, part F$?$

Consider the DE given at the start of this unit for a tapered column failing due to buckling:

$x^4\frac{d^{2}y}{d{x}^2}+\frac{4P}{c^{4}E}y=0$

A$.$ What are the functions $P(x)$ and $Q(x)$ when this is put in the standard form from

class? What type of point is $x_0=0?$

B$.$ Using the substitutions $t=\frac{1}{x}$ and $=\frac{4P}{c^{4}E},$ show that this equation may be rewritten

in this form:

$\frac{d^{2}y}{d{t}^2}+\frac{2}{t}\frac{dy}{dt}+y=0$

C$.$ What type of point is $t_0=0?$ What solution method can be used now, and what

minimal interval of convergence will it have?

D$.$ Find the general solution $y(t)$ using an appropriate series approach. If you have

the $C$ constant for the log term, you may assume it is equal to $0$ for this case $($for

reasons we won't cover in detail for this course$).$ Show all of your work.

E$.$ Substitute back in the original expressions for $x($but not $$ yet$)$ to produce $y(x),$

the general solution to the original ODE.

F$.$ Express $y(x)$ in terms of elementary functions by replacing each series solution,

then substitute $$ back out of the equation at the end. Your final answer should not

contain a summation sign ${\displaystyle \underset{?}{\overset{?}{}}},$ but it should be in terms of $x,P,c,$ and $E.($Hint: what

MacLaurin series are closest to your current series expressions?$)$