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Second Order Linear Homogeneous Differential Equations - from a younger Nerdi

This post is part of a series of older, now mostly-missing posts, in which I shared math/physics concepts I thought were cool. I was 14/15 when I made them (this was 2021), so a lot of them are not very good, but I'm reuploading them here for posterity's sake (plus, I don't post about physics nearly enough on here, for being a main focus of this site (>_>) I have also preserved their formatting as best as I can, even if that formatting isn't the best either.

second order linear homogeneous differential equations

ik its a mouthful but it really just means any equation like

ay''+by'+cy = 0 (using newton notation for derivatives)

we can find their solutions by asuming solution of form y = e^rx

then we get that

y´´= r^2 * e^(rx)

y´= r * e^(rx)

so we can rewrite the original eq like:

a * r^2 * e^(rx) + b * r * e^(rx) + c e^(rx) = 0

now, knowing that e^rx can't equal zero, we can factor it to say

ar^2 + br + c = 0

which is just a regular quadratic eq we can solve!

now, we can find solutions for r and substitute them into y = e^rx to find a solution to the second order linear homogeneous differential equation

note that i said: solutions. there is a more general form than these individual solutions but that is an exercise for the reader :troll: