In this section we will be looking exclusively at linear second order differential equations. The most common linear second order differential equation is in the type.
p (t ) y′′ + q (t) y′ + r (t ) y = g (t ) ....... (1)
Actually, we will hardly ever look at non-constant coefficient linear second order differential equations. In the section where we suppose constant coefficients we will use the subsequent differential equation.
ay′′ + by′ + cy+= g (t ) .... (2)
Where, probably we will utilize (1) only to make the point that specific facts, theorems, properties, or/and techniques can be used along with the non-constant form. Though, most of the time we will be using (2) as this can be fairly not easy to solve second order non-constant coefficient differential equations.
Firstly we will make our life easier through looking at differential equations along with g(t) = 0. As g(t) = 0 we call the differential equation homogeneous and as g (t ) ≠ 0 we call the differential equation non-homogeneous.
Therefore, let's start thinking about how to go regarding solving a constant homogeneous, coefficient, linear, second order differential equation. Now there is the general constant linear, coefficient, second order differential equation or homogeneous equation.
ay′′ + by′ + cy = 0
It's almost certainly best to start off with an illustration. This illustration will guide us to a very significant fact that we will use in every problem by this point on. The illustration will also provide us clues into how to go regarding to solving these in general.