Solve the differential equation using power series.

$(4x)y^{\text{'}\text{'}}3y^{\text{'}}5y=0,y(0)=1,y(0)=4$

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

You should get four power series:

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

Having $n=0$ in this power series yields a sero walue.

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

Find the recurrence relation.

Noten Cbserve that there are two answer boxess one is multiplied by $a_s$ and the other by $a_{s1}.$ Therefore, you only need to input expressions that multiply these coefficients feach input in the box should be an expression that depends only on the variable n$).$ Make sure to simplify each eupression as much as possible.

$a_{n2}=(,)*a_{n}1,-a_{51},n0$

Suppose we want to expand the series solution to diplay the first five terms, that is$,$ terms up to deyree four:

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

Find the corresponding coefficients.