<s>[INST] I'm stuck with this differential equations Mainly I need to get the real fundamental system of solutions of $y^{vi}+729y=0$.
As well, I'm stuck in getting the solution for $4y''+36y=\operatorname{cosc}(3x)$. Basically because I don't know how to operate it with the $\operatorname{cosc}(3x)$.
Thanks in advance.
 [/INST] For $y^{vi} + 729y = 0:$
Consider the characteristic equation $$c^6 + 729 = 0$$ or $$c^6 = -729$$
We can find the sixth roots of $-729$, and then convert those roots to the form $$c_n e^{\lambda t} \cos (\mu t) + c_{n+1} e^{\lambda t} \sin (\mu t)$$
To begin, let $z = -729 + 0i.$ then $|z| = |-729 + 0i| = 729$ and $\arg |z| = \arctan (\frac {0}{-729}) = \pi$.
Using DeMoivre's Theorem for roots $$z^{1/n} = |z|^{1/n} \left (\cos \frac {\theta k}{n} + i \sin \frac {\theta k}{n}\right)$$
we have $$729^{1/6} \left (\cos \frac {\pi k}{6} + i \sin \frac {\pi k}{6}\right)$$ which reduces to $$3 \left (\cos \frac {\pi k}{6} + i \sin \frac {\pi k}{6}\right)$$
Letting $k$ = $0$ through $5$ we get the following six roots:
$$\pm 3i, \pm \frac {3}{2} (\sqrt{3} + i), \pm \frac {3}{2} (1 + i \sqrt{3})$$
For each of the roots above, we can now use $c_{n} e^{\lambda t} \cos (\mu t) + c_{n+1} e^{\lambda t} \sin (\mu t)$ to find the solutions of the differential equation.
$3i: c_0 \cos 3t + c_1 \sin 3t$
$-3i: c_2 \cos 3t - c_3 \sin 3t$
(using $\lambda = 0, \mu = 3$)
$\frac {3}{2} (\sqrt{3} + i): c_4 e^{\frac {3t\sqrt {3}}{2}} \cos \frac {3t}{2} + c_5 e^{\frac {3t\sqrt {3}}{2}} \sin \frac {3t}{2}$
$\frac {3}{2} (\sqrt{3} - i): c_6 e^{\frac {3t\sqrt {3}}{2}} \cos \frac {3t}{2} - c_7 e^{\frac {3t\sqrt {3}}{2}} \sin \frac {3t}{2}$
(using $\lambda = \frac {3\sqrt {3}}{2}, \mu = \frac {3}{2}$)
$\frac {3}{2} (1 + i\sqrt{3}): c_8 e^{\frac {3t}{2}} \cos \frac {3\sqrt{3}}{2} t + c_{9} e^{\frac {{3t}}{2}} \sin \frac {3\sqrt{3}}{2} t$
$\frac {3}{2} (1 - i\sqrt{3}): c_{10} e^{\frac {3t}{2}} \cos \frac {3\sqrt{3}}{2} t - c_{11} e^{\frac {{3t}}{2}} \sin \frac {3\sqrt{3}}{2} t$
(using $\lambda = \frac {3}{2}, \mu = \frac {3\sqrt {3}}{2}$)
The solution is then
$$(K_0) \cos 3t + (K_1) \sin 3t + (K_2)e^{\frac {3t\sqrt {3}}{2}} \cos \frac {3t}{2} + (K_3) e^{\frac {3t\sqrt {3}}{2}} \sin \frac {3t}{2} + (K_4) e^{\frac {3t}{2}} \cos \frac {3\sqrt{3}}{2} t + (K_5) e^{\frac {3t}{2}} \sin \frac {3\sqrt{3}}{2} t$$ 
where 
$$\begin{matrix} 
K_0 = (c_0 + c_2) \\
K_1 = (c_1 - c_3) \\
K_2 = (c_4 + c_6) \\
K_3 = (c_5 - c_7) \\
K_4 = (c_8 + c_{10})\\
K_5 = (c_{9} - c_{11})\\
\end{matrix}$$
</s>