Differential Equation's: A popular solving method

1. Oct 9, 2007

AbedeuS

Hey, I've recently enrolled in a maths module involving alot of differential equations, most of the material is all fine and dandy, but I get somewhat confused at this method of solving them.

Scenario

Solve:

$$\frac{d^{2}y}{dx{2}} + 2\frac{dy}{dx} + 3y$$

So then theres this technique we were taught to use, where we define the initial function as just $$e^{mx}$$ then differentiate and substitute and stuff, as such:

$$\frac{d^{2}e^{mx}}{dx^{2}}+2\frac{d e^{mx}}{dx}+3e^{mx}$$

To give:
$$m^{2}e^{mx}+2me^{mx}+3e^{mx}$$

Then:
$$e^{mx}(m^{2}+2m+3)$$ and so on and so fourth through quadratic equations and such.

But what am I technically doing, am i just "Guessing" that the origional function was an exponential with a coefficiant on the power, rather than the "True" form of the function instead of just differentiating the origional function?

Or is this method usually employed when the initial function is not known (or is too complex) and the exponential method is just a nice way of dealing with it? Does it always work?

Thanks
-Abe