Example 6. Given the general solution $y=C_{1} \sin 2 x+C_{2} \cos 2 x$ of the differential equation $y^{\prime \prime}+4 y=0$. What particular solutions are obtained when $C_{1}=2, C_{2}=3$? For what values of the parameters $C_{1}$ and $C_{2}$ do the particular solutions $y=\sin 2 x, y=\cos 2 x$ result?
Solution. Substituting the values $C_{1}=2, C_{2}=3$ into the formula for the general solution, we obtain the particular solution $y=2 \sin 2 x+3 \cos 2 x$. The particular solution $y=\sin 2 x$ is obtained from the general solution when $C_{1}=1, C_{2}=0$, and the particular solution $y=\cos 2 x$ when $C_{1}=0, C_{2}=1$.