If y1 (t) and y2 (t) are two solutions to a linear, homogeneous differential equation thus it is y (t ) = c1 y1 (t ) + c2 y2 (t ) ........................(3)
Remember that we didn't comprise the restriction of constant coefficient or second order in this. It will work for any linear homogeneous differential equation.
If we further suppose second order and one other condition that we'll provide in a second we can go a step further.
If y1 (t) and y2 (t) are two solutions to a linear, second order homogeneous differential equation and they are "nice enough" so the general solution to the linear, second order differential equation is specified by (3).
So, just what do we mean by "nice enough"? We'll hold off on that until a later section. At this point you'll hopefully believe it when we say that specific functions are "nice enough".
Thus, if we now make the assumption as we are dealing along with a linear, second order differential equations, we now identify that (3) will be its general solution. The subsequent question which we can ask is how to get the constants c1 and c2. Because we have two constants it makes sense, confidently, which we will require two equations or conditions to get them.
One manner to do this is to identify the value of the solution at two distinct points or
y (t0) = y0
y (t1) = y1
These are usually termed as boundary values and are not actually the focus of this course thus we won't be working along with them.
The other way to get the constants would be to identify the value of the solution and its derivative at an exacting point. Or,
y (t0) = y0
y′ (t0) = y0′
These are the two conditions which we'll be using here. When with the first order differential equations these will be termed as initial conditions.