= Solution
The <homogeneous solution> is
$$
\Delta y_{\rm h}=A\cos\phi+B\sin\phi.
$$
For the constant and $\cos2\phi$ forcing terms, a <particular solution> is
$$
\Delta y_{\rm p}
=\frac{3M}{2b^2}+\frac{M}{2b^2}\cos2\phi.
$$
Indeed, applying $d^2/d\phi^2+1$ multiplies the $\cos2\phi$ coefficient by $-3$. Thus
$$
\boxed{
\Delta y
=A\cos\phi+B\sin\phi
+\frac{M}{2b^2}(3+\cos2\phi).}
$$
The $B\sin\phi$ term only changes the definition of $b$ at this order, so one may set $B=0$.
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