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

[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?

 

[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.