2. Find the general solution of $$ \frac{d^{2} y}{d t^{2}}+18 y=0 . $$

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1. We start with the differential equation
$$
\frac{d^2 y}{d t^2} + 18y = 0.
$$

2. Assume a solution of the form
$$
y(t) = e^{rt},
$$
where $$ r $$ is a constant. Substituting into the differential equation gives
$$
r^2 e^{rt} + 18 e^{rt} = 0.
$$

3. Since $$ e^{rt} \neq 0 $$ for any $$ t $$, we divide by $$ e^{rt} $$ to obtain the characteristic equation:
$$
r^2 + 18 = 0.
$$

4. Solve the characteristic equation for $$ r $$:
$$
r^2 = -18 \quad \Rightarrow \quad r = \pm i\sqrt{18}.
$$
Simplify $$ \sqrt{18} $$:
$$
\sqrt{18} = 3\sqrt{2}.
$$
Thus,
$$
r = \pm 3i\sqrt{2}.
$$

5. The general solution for a second order linear homogeneous differential equation with complex conjugate roots $$ r = \pm i\omega $$ is given by:
$$
y(t) = C_1 \cos(\omega t) + C_2 \sin(\omega t),
$$
where $$ \omega = 3\sqrt{2} $$ and $$ C_1, C_2 $$ are arbitrary constants.

6. Therefore, the general solution is:
$$
y(t) = C_1 \cos(3\sqrt{2}\, t) + C_2 \sin(3\sqrt{2}\, t).
$$
The general solution is $$ y(t) = C_1 \cos(3\sqrt{2}\, t) + C_2 \sin(3\sqrt{2}\, t) $$.

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