Good day everyone,
Studying for the first time 2nd order differential equations, we focused on the case of linear homogeneous equations. However, it comes out that the general integral, or general solution (or structure theorem) of the equation, well this theorem, can be proved in more advanced courses.
Given AY''+BY' +CY=0 where A, B, C are reals and A is different from 0,the general solution is the linear combination of two linearly independent solutions Y(x) = C1*Y1+C2*Y2, where C1 and C2 are arbitrary numbers.
However, it turns out as well that it is possible to give an elementary demonstration by constructing Cauchy problems and demonstrating that any particular solution can be represented as a linear combination of 2 solutions with specific coefficients.
Could someone please explain to me the intuition which lies behind this demonstration and its steps?
Thank you in advance!