Suppose I have a third order differential equation, and have three solutions, y1, y2, y3.(adsbygoogle = window.adsbygoogle || []).push({});

I can check to see if they are linearly independent as such: if their wronskian is non-zero, they are linearly independent.

But the wronskian is just a determinant of a matrix.

y1 y2 y3

y1' y2' y3'

y1'' y2'' y3''

This is extremely analogous to vectors, because if I have two vectors in some based system (e1,e2,e3), I can see if these two vectors are linearly independent by taking their cross product. If the cross product is non-zero, then the vectors are linearly independent.

But the cross product is just the determinant of a matrix as well.

e1 e2 e3

a1 a2 a3

b1 b2 b3

So if I'm doing the exact same thing, these two ideas must be related, no?

If we let some based system be (y1,y2,y3) (the idea of having a system based on the functions seems weird and kind of circular, for in order for them to form a basis they have to be linearly independent in the first place, but I'll keep typing anyway..)

Then showing that the vectors

(y1',y2',y3') and (y1'', y2'', y3'') are linearly independent becomes the exact same task as showing that y1, y2, and y3 are linearly independent solutions to a differential equation.

Why?

(I realize that it could be the case that the two "linearly independent" labels are not related for vectors and solutions to DEs in any way, but I chose to assume that there must be some correlation.)

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# Differential Equations and a vector analogy (weird question)

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