I am reading on the solutions of second order homogenous ODEs (linear), and came across this:
Now, my questions is split into two parts:
1) I know that the independent solutions are equations, however are they solutions to same differential equation (I mean to say, "are they both solutions derived from the second integration"), or for example, is $y_1$ the solution of the first integration and $y_2$ the solution of the second?
2)Now, if these independent solutions are indeed the solutions to the same differential equations, or the above example is true, or whatever else the independent solutions resemble, why is it that their sum is also a solution?
Best Answer
The two independent solutions are functions, not equations. The functions can (sometimes) be defined by equations. Both functions are solutions of the same equation. This is very much analogous to having two solutions $x$ to $x^2-1=0$.
If $y_1$ and $y_2$ are functions $\mathbb R\to\mathbb R$, so is $y_1+y_2$. This function is defined via $(y_1+y_2)(t)=y_1(t)+y_2(t)$. Checking that $y_1+y_2$ also solves the differential equation is a calculation that I urge you to do yourself. (The underlying reason is that the equation is linear, but you need not worry about that here. You can just calculate.)