Rice: Created page with "Category:Differential Equations 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 $L[y]=y''+py'+qy$ It can be (pretty easily) proven that $L[y_1+y_2]=L[y_1]+L[y_2]$ and $L[cy]=cL[y]$ Then, given two homogeneous solutions $L[y_1]=L[y_2]=..."$

2024-05-18T00:34:09Z

Created page with "Category:Differential Equations 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 <math> L[y]=y''+py'+qy </math> It can be (pretty easily) proven that <math> L[y_1+y_2]=L[y_1]+L[y_2] </math> and <math> L[cy]=cL[y] </math> Then, given two homogeneous solutions <math> L[y_1]=L[y_2]=..."

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[[Category:Differential Equations]]

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

<math>
L[y]=y''+py'+qy
</math>

It can be (pretty easily) proven that

<math>
L[y_1+y_2]=L[y_1]+L[y_2]
</math>

and

<math>
L[cy]=cL[y]
</math>

Then, given two homogeneous solutions

<math>
L[y_1]=L[y_2]=0
</math>

We can prove that

<math>
L[c_1y_1+c_2y_2]=0
</math>

And by extension, we can also find the solutions of nonhomogeneous solutions with one particular solution and the two homogeneous solutions.

Superposition principle - Revision history