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.
Let's introduce a bit of spice. Express y as
Now, differentiate y twice.
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
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
If the discriminant is negative, the solution to this quadratic will be a complex conjugate pair,