Find the general solutions of the following equations and compare computer solutions.

$y^{\text{'}\text{'}\text{'}}+y=0$

Given equation is $y^{\text{'}\text{'}\text{'}}+y=0$.

A differential equation is a formula that connects the derivatives of one or more unknown functions. Functions are used to represent physical quantities, derivatives are used to characterize their rates of change, and differential equations are used to define a relationship between them in applications.

The auxiliary equation is

$$\begin{gathered} \left(D^3+1\right)y=0 \\ (D+1)\left(D^2-D+1\right)y=0 \\ (D+1)\left(D-\frac{1+i\sqrt{3}}{2}\right)\left(D-\frac{1+i\sqrt{3}}{2}\right)y=0 \end{gathered}$$

It is found that the general solution for any differential that has an auxiliary equation with unequal roots is

$y=c_1{e}^{a_{1}x}+c_2{e}^{a_{2}x}+\dots ..+c_n{e}^{a_{n}x}$

But here there are two complex roots here the general solutions would be a linear combination between the three solutions of each part of the auxiliary equation $a_1=-i,a_2=i,a_3=-1,a_2=1$. Then, the differential equation's generic solution is

$y=c_1{e}^{-x}+e^{x/2}\left[c_2\sin \left(\frac{\sqrt{3}}{2}x\right)+c_3\cos \left(\frac{\sqrt{3}}{2}x\right)\right]$