Differential Equations - Second Order

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Homework Statement

Hi,

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

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

Homework Equations

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.

 

[1MileCrash]

That equation is homogeneous, you only need to assume that V=e^rt, no need for the A, it would just be merged with an arbitrary constant. You then plug that into the DE, getting you what we call a characteristic equation (which will be a simple quadratic of r, I think you can do it easily.) Also, "assuming" is mathematical. Because that's a great assumption. A function, plus its derivative, plus its second derivative, being zero, must be some function with a repetitive differentiation pattern. It's not just an "assume this because I say so." We make assumptions of the form of an equation and deduce the specifics of our assumption for the answer, all the time. Doing so is by far and wide the cleanest, most elegant way to solve this problem.