Problem #$3$: Consider the following differential equation.

$(13x^2)y^{\text{'}\text{'}}-2x{y}^{\text{'}}-12y=0$

$($a$)$ If you were to look for a power series solution about $x_0-0,$ i$.$e$.,$ of the form

${\displaystyle \underset{n=0}{\overset{}{}}}{c}_n{x}^n$

then the recurrence formula for the coefficients would be given by $c_{k2}-g(k)c_k,k2.$ Enter the function $g(k)$ into the answer box below.

$($b$)$ Find the solution to the above differential equation with initial conditions $y(0)=0$ and $y^{\text{'}}(0)=4.$

$($Note that this solution is a terminating power series.$)$

$($c$)$ Find the first three nonzero terms in the solution to the above differential equation with initial conditions $y(0)=6$ and $y^{\text{'}}(0)=0.$

Problem #$3($a$)$: $$

Enter your answer as a symbolic

function of $k,$ as in these

examples

Protilem $23($b$)$: $$

Enter your answer as a symbolic

function of $x_1$ as in these

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Problem $73(c)$ : $$

Enter your answer as a symbolic

function of $x,$ as in these

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