= Solution
For real parameters, work on $x>0$ so arbitrary real powers are unambiguous. Taking <derivatives> gives
$$
y_1'=-\gamma x^{\gamma-1}\sin(x^\gamma),\qquad y_1''=-\gamma(\gamma-1)x^{\gamma-2}\sin(x^\gamma)-\gamma^2x^{2\gamma-2}\cos(x^\gamma).
$$
The residual after substitution is
$$
\bigl(1-a\gamma^2x^{\alpha+2\gamma-2}\bigr)\cos(x^\gamma)-\gamma\bigl(a(\gamma-1)+b\bigr)x^{\alpha+\gamma-2}\sin(x^\gamma).
$$
When $\gamma\ne0$, $x^\gamma$ ranges over $(0,\infty)$. Evaluating at the zeros of the cosine first forces the sine coefficient to vanish; then the cosine coefficient vanishes identically. If $\gamma=0$, the nonzero constant $\cos1$ cannot satisfy the equation. Thus the necessary and sufficient conditions are
$$
\boxed{\gamma\ne0,\qquad \alpha=2-2\gamma,\qquad a=\gamma^{-2},\qquad b=(1-\gamma)\gamma^{-2}.}
$$
Under these conditions the normalized coefficient is $p=(1-\gamma)/x$. The <Abel identity> gives $W=Cx^{\gamma-1}$. Choose $C=\gamma$ and apply <reduction of order> on an interval where $\cos(x^\gamma)\ne0$:
$$
y_2=\cos(x^\gamma)\int\frac{\gamma x^{\gamma-1}}{\cos^2(x^\gamma)}\,dx=\cos(x^\gamma)\tan(x^\gamma).
$$
The substitution $u=x^\gamma$ evaluates the <integral>. After continuation across the cosine zeros,
$$
\boxed{y_2(x)=\sin(x^\gamma),\qquad W(y_1,y_2)=\gamma x^{\gamma-1}\ne0.}
$$
This is a <power substitution for an oscillatory second-order equation>: in the variable $u=x^\gamma$, the equation reduces to the <harmonic oscillator equation>.
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