Why do solutions to ODE need linear independence?

T

ted s

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since the solutions will be of the form c1 g(x) + c2 w(x) where both g(x) and w(x) solve the ODE if w(x) = c3 g(x) then you really only have [c1 + c3] g(x) = C4 g(x)...one solution
 
since the solutions will be of the form c1 g(x) + c2 w(x) where both g(x) and w(x) solve the ODE if w(x) = c3 g(x) then you really only have [c1 + c3] g(x) = C4 g(x)...one solution
 
since the solutions will be of the form c1 g(x) + c2 w(x) where both g(x) and w(x) solve the ODE if w(x) = c3 g(x) then you really only have [c1 + c3] g(x) = C4 g(x)...one solution
 
since the solutions will be of the form c1 g(x) + c2 w(x) where both g(x) and w(x) solve the ODE if w(x) = c3 g(x) then you really only have [c1 + c3] g(x) = C4 g(x)...one solution
 
I thought about this some and it seems that solutions to second order linear ordinary differential equations (Im just trying to sort this all out) need to contain two linear independent solutions. Why is this? What happens if the solutions are linear dependent?
 
since the solutions will be of the form c1 g(x) + c2 w(x) where both g(x) and w(x) solve the ODE if w(x) = c3 g(x) then you really only have [c1 + c3] g(x) = C4 g(x)...one solution
 
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