= Solution
The associated <homogeneous linear differential equation> has <characteristic roots> $2\pm i$, so its <complementary function> is $e^{2x}(A\cos x+B\sin x)$. For a <particular integral>, the <method of undetermined coefficients> suggests $y_p=ax^2+bx+c$. Substitution and comparison of <coefficients> gives
$$
5a=125,\qquad -8a+5b=0,\qquad 2a-4b+5c=0,
$$
so $y_p=25x^2+40x+22$. The <general solution> is therefore
$$
y=25x^2+40x+22+e^{2x}(A\cos x+B\sin x).
$$
The first <boundary condition> gives $A=-21$. At $x=\pi/2$, the polynomial part already equals the assigned boundary value and the cosine vanishes, so $e^\pi B=0$. Thus
$$
\boxed{y(x)=25x^2+40x+22-21e^{2x}\cos x}.
$$
This is the unique solution satisfying the two <boundary conditions>; here the phrase “particular solution” means that selected member of the full family, rather than only the polynomial <particular integral>.
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