# 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 roots`m1` and `m2` are real and distinct, the general solution is `y = Ae^(m1x) + Be^(m2x)` where `A` and `B` are arbitrary constants.
• If the roots`m1` 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.