The general solution to $$(d^2y)/(dt^2)+3(dy)/dt+5y=0$$ is:

This is a 2nd-order linear homogenous differential equation with constant coefficients. It takes the form of:

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

For these types of differential equations, we first want to find the characteristic equation, which resembles the following equation:

$r^2+ar+b=0$ 

As we can see, the characteristic equation is a quadratic equation. Depending on the roots for this equation, it will give us the form of our general solution. These are listed on pg. 52 of our FE Reference Handbook:

Let's now write the characteristic equation of our problem:

$$\begin{gathered} r^2+ar+b=0 \\ {r}^2+3r+5=0 \\ a=3, b=5 \end{gathered}$$

Since $(a^2 = 3^2 = 9)< (4b = 4·5 = 20)$, our general solution must take the form:

$$\begin{gathered} y = e^{\alpha x}(C_1\cos \beta x+C_2\sin \beta x) where \\ \alpha = -a/2 \\ \beta = \frac{\sqrt{4b-a^2}}{2} \end{gathered}$$

Solving for alpha and beta and plugging them into our general solution:

$$\begin{gathered} \alpha = -3/2 \\ \beta =\frac{\sqrt{4\left(5\right)-\left(3^2\right)}}{2}=\frac{\sqrt{11}}{2} \end{gathered}$$

$y = e^{\frac{-3}{2}\mathrm{x}}\left(C_{1}cos\right(\frac{\sqrt{11}\mathrm{x}}{2})+C_{2}sin(\frac{\sqrt{11}\mathrm{x}}{2}\left)\right)$

Therefore, this is our general solution, where $C_{1 }$ and $C_2$ are arbitrary constants.