$9$ points

Using Power series solution for a second order homogeneous differential equation around $x=0,$ we get $c_{k+2}=\frac{1}{(k+1)(k+2)}{c}_k$ for $k=0,1,2,$cdots.

Then the general solution is $y=c_0{y}_1+c_1{y}_2$ with:

$y_1=1+\frac{x^2}{2}+\frac{x^4}{24}+\frac{x^6}{720}+$cdots

$$ None of the mentioned $$ The above option

$y_1=x+\frac{x^3}{2}+\frac{x^5}{24}+\frac{x^7}{720}+$cdots,$y_1=1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+$cdots

The above option $$ The above option