Solution
= Solution
The <characteristic roots> are
$$
m=\frac{-3\pm\sqrt{9-16}}2=-\frac32\pm i\frac{\sqrt7}{2}.
$$
Taking real <linear combinations> of the two <complex exponential> solutions yields
$$
\boxed{y(x)=e^{-3x/2}\left[C_1\cos\left(\frac{\sqrt7}{2}x\right)
+C_2\sin\left(\frac{\sqrt7}{2}x\right)\right]}.
$$
The coefficient $\beta$ is three here, as confirmed in the PDF; the OCR replaces it by an undefined symbol.