Solution
= Solution
The <characteristic equation> of the constant-coefficient <linear ordinary differential equation> is
$$
4r^2+8r+5=0,
$$
whose <roots of a polynomial> are $r=-1\pm i/2$. Taking real and imaginary parts of the corresponding <exponential functions> gives the real general solution
$$
\boxed{y_h(x)=e^{-x}\left(A\cos\frac x2+B\sin\frac x2\right),\qquad A,B\in\mathbb R.}
$$