11 - Complex Numbers and Quadratics in Solving Second Order Linear Differential Equations

So, we can, in a sense, use a form of quadratics to solve second order differential equations with constant coefficients. This means that on the derivatives we have no other variables, only constants.

Now, we know of one function that repeats itself when differentiating:

A little change: r

Let's introduce a bit of spice. Express y as

Now, differentiate y twice.

Express this differential equation in these new terms

Since can never be zero,

We can now treat this as a quadratic equation. We'll have three cases.

Case 1: D > 0

If the discriminant is larger than zero, there will be two real solutions to this quadratic, and the general solution to our differential equation will be of form

Case 2: D = 0

If the discriminant is zero, there will be one real solution to this quadratic, and the general solution to our differential equation will be of form

Case 3: D < 0

If the discriminant is negative, the solution to this quadratic will be a complex conjugate pair, and , and the general solution to our differential equation will be of form