Why does the 2nd order homogeneous linear ODE have 2 general solutions?

1. Dec 11, 2014

kidsasd987

why not the 2nd order linear homogeneous ODEs have three Linearly independent solutions or more? I know for the characteristic equation, we can only find 2 answers but.. just wondering if that is the only case to solve the question and if it is, then why it has to be.

so my question is,

1. 2nd order linear homogeneous ODE has 2 general solutions. but why?

2. derivation?

Last edited: Dec 11, 2014
2. Dec 11, 2014

kidsasd987

is that because all 2nd order linear homogeneous ODEs can be solved by characteristic equation?

so the logic flow here is, 1. all 2nd order linear homogeneous ODES can be solved by characteristic eqs. 2. therefore we have 2 solutions.

not the reverse.