Second Order Homogeneous Differential Equations

Definition and Form

A second order homogeneous differential equation is an equation of the form ay'' + by' + cy = 0, where a, b, and c are constants and ' denotes a derivative.
The term ‘second order’ refers to the fact that the equation involves the second derivative of a function.
The term ‘homogeneous’ means that the equation is set equal to zero.

Characteristic Equation

The characteristic equation of the differential equation can be obtained by setting y = e^(mx) for some constant m, and substituting into the differential equation to get am^2 + bm + c = 0.
This quadratic equation gives two roots, which can be real and distinct, real and the same, or complex.

Solution Types

The form of the general solution depends on the roots of the characteristic equation.
If the rootsm1 and m2 are real and distinct, the general solution is y = Ae^(m1x) + Be^(m2x) where A and B are arbitrary constants.
If the rootsm1 and m2 are real and the same (that is, m1 = m2 = m), the general solution is y = (Ax + B)e^(mx) where A and B are arbitrary constants.
If the roots m1 and m2 are complex conjugates (that is, m1 = p + qi and m2 = p - qi), the general solution is y = e^(px)(Acos(qx) + Bsin(qx)) where A and B are arbitrary constants, and i is the imaginary unit.

Remember, solving differential equations often requires careful casework depending on the roots of the characteristic equation. Equipped with this understanding, you are well prepared to tackle problems involving second order homogeneous differential equations.