Why does reduction of order work for linear ODEs?

[theBEAST]

This is not a homework problem, I just want to understand some theory behind this mathematical method.

Specifically, if we know that one solution is y₁(t), then why is the second solution in the form

y₂(t) = v(t) * y₁(t)?​

Where v(t) is the function that you need to solve for. Why does this assumption always work?

 

[SammyS]

This is not a homework problem, I just want to understand some theory behind this mathematical method.

Specifically, if we know that one solution is y₁(t), then why is the second solution in the form

y₂(t) = v(t) * y₁(t)?​

Where v(t) is the function that you need to solve for. Why does this assumption always work?

The Wikipedia entry for reduction of order gives a good explanation.

http://en.wikipedia.org/wiki/Reduction_of_order