= Solution
Use a <power-series solution of a differential equation>, $y=\sum_{n\geq0}c_nx^n$. Comparing coefficients gives
$$
(n-1)(n-2)c_n=c_{n-2},\qquad c_{-1}=c_{-2}=0.
$$
In particular $c_0=0$, while $c_1$ and $c_2$ are free. The two prescribed derivative pairs select $(c_1,c_2)=(a,0)$ and $(0,b)$. The next coefficients are $c_3=c_1/2$, $c_4=c_2/6$, $c_5=c_1/24$, $c_6=c_2/120$. Hence
$$
\boxed{y_1(x)=a\left(x+\frac{x^3}{2}+\frac{x^5}{24}+O(x^7)\right)},\qquad
\boxed{y_2(x)=b\left(x^2+\frac{x^4}{6}+\frac{x^6}{120}+O(x^8)\right)}.
$$
These are the first three nonzero terms when the respective parameter is nonzero; if $a=0$ or $b=0$, the corresponding solution vanishes identically.
For closed forms, set $y=xu$. Cancellation of the first-derivative terms reduces the equation, for $x\ne0$, to $u''-u=0$. The analytic solutions extend across zero, and their <hyperbolic cosine> and <hyperbolic sine> expansions identify
$$
\boxed{y_1=ax\cosh x,\qquad y_2=bx\sinh x}.
$$
This is the <hyperbolic reduction of a regular-singular differential equation>: although the original leading coefficient vanishes at zero, both selected <power-series solutions of a differential equation> are entire.
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