I explained that in general one expects to index solutions to a second order equation by two constants of integration; for example the solutions to y"=0 are ax+b. Initial conditions will often consist of giving both y and y' at a point. You don't get a direction field any more.
Second order equations are important in physics because of F=ma. I set up the central example that we will return to repeatedly for insight: the spring-mass-dashpot system. Take x=0 as rest position. Then the spring exerts a restoring force given by F_spring(x,t), where F'(x,t)<0. The dashpot exerts a restraining force, depending (in a sufficiently homogeneous situation) only on v: F_dash(v,t). There may also be an external force F_ext. A further simplifying assumption is that F_ext depends onlly on t. Then
m(t)x" - F_dash(x',t) - F_spring(x,t) = F_ext(t)
The LHS represents the system, the RHS the impressed force, as before.
For x and x' reasonably small we can approximate F_dash(x')=-cx' and F_spring(x)=-kx. This is the LINEAR assumption on the system, and it leads to
m(t)x" + c(t)x' + k(t)x = F_ext(t)
This is an example of a second order linear equation, ie it can be put in the "standard form"
y" + p(x)y' +q(x)y = r(x) (*)
For example y" + y = 0 has cos(x) and sin(x) as solutions, and more generally a cos(x) + b sin(x) is a solution.
(*) has "constant coefficients" if p and q are constant. (*) is "homogeneous" if r=0: the "free system."
y" + p(x)y' + q(x) = 0 (H)
is the "homogeneous equation associated to (*)."
General comments:
(1) If y1 and y2 are solutions to (H) then so is a y1 + b y2.
(2) If yh is a solution to (H) and yp is a solution to (*) then yp + yh is a solution to (*).
For example, x is a solution to y" + y = x, so x + a cos(x) + b sin(x) is too.
The set of solutions of (H) is analogous to a plane through the origin. The set of solutions of (*) is analogous to a plane not (necessarily) through the origin.
Definition: y1 and y2 are "linearly independent" if neither is a multiple of the other. Equivalently, a y1 + b y2 = 0 forces a=b=0.
For example, cos(x) and sin(x) are linearly independent, since if a cos(x) + b sin(x) = 0 then taking x=0 forces a=0 while taking x=pi/2 forces b=0.
Theorem: (*) admits a solution yp. (H) admits a linearly independent pair of solutions y1, y2. For any choice of yp, y1, y2, the general solution to (*) is yp + ay1 + by2.