Nature Of Solutions For Second Order Linear Homogeneous Differential Equations

Given that ay"+by'+cy=0 is a second order Linear Homogeneous Differential Equation To determine it's solution, we want to transform it in to an auxiliary equation or the characteristics equation in the form: ar²+br+c=0, which is a Quadratic Equation that we need to to solve for r.
From the general formula for solving Quadratic equation: r = (-b±√D)/2a where
D = b²-4ac

To determine the form of Solution, we have to consider the nature of our roots:
if D greater than 0 we have real and distinct roots, and therefore the general solution will be in the form:
y=C₁eʳ¹ˣ+C₂eʳ²ˣ.

But what happens if D=0? Well then instead we use: y=(C₁+C₂x)eʳˣ because we have real and equal roots.

Additionally, it’s important to realise that our r may not necessarily be real numbers. If they happen to be complex, we could call our two solutions r1=m+sι and r2=m-sι, since they’ll always be complex conjugate pairs. Then our solution for y, using the relations between eit and the trigonometric functions, can be written as: y=eᵐˣ[C₁cos(st)+C₂sin(st)].

So these three formula we’ve ended up with are all we actually need to remember. For any homogeneous second order differential equation with constant coefficients, we simply jump to the auxiliary equation, find our r, write down the implied solution for y and then use initial conditions to help us find the constants if required.

[georgesadler]

Professor Tambuwal thank you for a short video/lecture on Solution to Second Order Linear Homogeneous Differential Equations.