Differential Equations - Second Order

Homework Statement

Hi,

It's been a while since I have taken differential equations. How do I solve an equation like this:

$k_{1}\frac{d^{2}V_{x}(t)}{dt^{2}}+k_{2}\frac{dV_{x}(t)}{dt}+k_{3}V_{x}(t)=0$

The Attempt at a Solution

I have looked through my books and it says to assume that

$V_{x}(t) = Ae^{st}$

The book then goes on to solve the equation which I can follow. However I would like to know how to actually solve the problem mathematically without "assuming". If I remember correctly you can use something called Laplace transforms. However, I don't really remember if you can solve this equation using Laplace transforms. Thanks for any help.