Answer
$$
y^{\prime \prime}+y=0, \quad y(\pi / 3)=2, \quad y^{\prime}(\pi / 3)=-4
$$
The general solution of the given initial value problem is
$$ y(t) =(1+2\sqrt{3} ) \cos t+ (\sqrt{3}-2 ) \sin t $$
$ y(t) $ has steady oscillation.
Work Step by Step
$$
y^{\prime \prime}+y=0, \quad y(\pi / 3)=2, \quad y^{\prime}(\pi / 3)=-4
\quad (1)
$$
We assume that $ y = e^{rt}$, and it then follows that $r$ must be a root of the characteristic equation
$$
r^{2}+1=0, $$
so its roots are
$$
\:r_{1,\:2}=\frac{ \pm \sqrt{-4\cdot \:1\cdot \:1}}{2\cdot \:1}
$$
Thus the possible values of $r$ are
$$r_{1}= i ,\quad r_{2}=-i .$$
Therefore two solutions of Eq. (1) are
$$
y_{1}(t) = e^{(i )t}=\cos t + i \sin t
$$
and
$$
y_{2}(t) = e^{(-i )t}=\cos t - i \sin t
$$
Thus the general solution of the differential equation is
$$
y(t) =c_{1} \cos t+c_{2} \sin t \quad\quad\quad (2)
$$
where $ c_{1} $ and $c_{2}$ are arbitrary constants.
To apply the first initial condition, we set $t = \pi / 3$ in Eq. (ii); this gives
$$
\begin{split}
y( \pi / 3) & = c_{1} \cos \pi / 3+c_{2} \sin \pi / 3 = 2 \\
& = \frac{1}{2} . c_{1} + \frac{\sqrt{3}}{2} . c_{2} = 2 \quad \quad\quad (3)
\end{split}
$$
For the second initial condition we must differentiate Eq. (2) as follows
$$
y^{\prime}(t) =c_{2} \cos t - c_{1} \sin t
$$
and then set $t =\pi / 3 $. In this way we find that
$$
\begin{split}
y^{\prime}(\pi / 3 ) & =c_{2} \cos t - c_{1} \sin t
=-4 \\
& = c_{2} \cos (\pi / 3) - c_{1} \sin (\pi / 3)
=-4 \\
&=\frac{1}{2} . c_{2} - \frac{\sqrt{3}}{2} .c_{1}
=-4 \quad \quad \quad \quad (4)
\end{split}
$$
from eq.(3) and eq.(4), it follows that
$$
\left\{\begin{array}{ll}{\frac{1}{2} . c_{1} + \frac{\sqrt{3}}{2} . c_{2} } & {=2} \\ {\frac{1}{2} . c_{2} - \frac{\sqrt{3}}{2} .c_{1}
} & {=-4}\end{array}\right.
$$
This system of linear equations evaluates to
$$ c_{1} = 1+2\sqrt{3} , \quad\quad c_{2} = \sqrt{3}-2 $$.
Using these values of $ c_{1} = 1+2\sqrt{3} , c_{2} = \sqrt{3}-2 $, in Eq. (2), we obtain
$$
y(t) =(1+2\sqrt{3} ) \cos t+ (\sqrt{3}-2 ) \sin t
$$
as the solution of the initial value problem (1).
