Given the second order homogenous differential equation

Question

anonymous · Answer

$$y" +\alpha y'+\beta y =o$$
let sum of the root is 94 and the product is 2013. solve the DE

anonymous · Answer

The characteristic equation you get from the DE is
$$r^2+\alpha r+\beta=0$$
You know that the sum of the roots to this equation is 94, and that their product is 2013. So, if $r_1$ and $r_2$ are roots, then
$$\begin{cases}
r_1+r_2=94\
r_1r_2=2013
\end{cases}$$
Solving by substitution works nicely:
$$\begin{align*}r_2=94-r_1~\Rightarrow~&r_1(94-r_1)=2013\
&r_1^2-94r_1+2013=0\
&r_1=33~\text{or}~r_1=61
\end{align*}$$
If $r_1=33$ then $r_2=61$. Solving for $\alpha$ and $\beta$ is pretty simple: just sub these roots for $r$ and solve the system of equations. From there the solution to the DE is easy to find.