Solve the differential equation using power series.

$y^{\text{'}\text{'}}-5x{y}^{\text{'}}-5y=0,y(0)=4,y^{\text{'}}(0)=2$

Express each term of the differential equation as a power series. Shift indices, where necessary.

Why are we allowed to make the second power series start from $n=0?$

In any power series the term with $n=0$ equals sere.

Having $n=0$ in this power series yields a zero value.

Find the recurrence relation. Note: Observe that the answer is in units of $a_a,30$ you only need to input an expression that multiplies $a_n($therefore$,$ your input should be an expression that depends only on the variable n$).$

$a_{s42}=,-a_s,$ where $n0$

Suppose we want to expand the series solution to display the first five terms, that is$,$ want to present terms up to degree four.

$y=a_0{a}_{1}x{a}_2{x}^2{a}_3{x}^3{a}_4{x}^4$ dots

Find the corresponding coefficients.

$a_0=$

$a_1=$

$a_2=$