Superposition principle
From Rice Wiki
The superposition principle is actually a broad category of principles that widely apply to homogeneous linear stuff. I'll discuss this in the context of differential equations.
Consider a linear operator on functions, L
![{\displaystyle L[y]=y''+py'+qy}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19921f12876bced6b83889a3f54bc9a5c45a379c)
It can be (pretty easily) proven that
![{\displaystyle L[y_{1}+y_{2}]=L[y_{1}]+L[y_{2}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d19dd57a828d047e1a34c29dd88ab7269e53cfb)
and
![{\displaystyle L[cy]=cL[y]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d212bdd9ffbe58ae0eb7e770a4ec817a581f28f6)
Then, given two homogeneous solutions
![{\displaystyle L[y_{1}]=L[y_{2}]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e2cb21d103ca92d103b257b1769e0b7c1a3096f)
We can prove that
![{\displaystyle L[c_{1}y_{1}+c_{2}y_{2}]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4df533bbd650a00ef75f888209d4da84b9a6b426)
And by extension, we can also find the solutions of nonhomogeneous solutions with one particular solution and the two homogeneous solutions.
