Linear differential equations with constant coefficients

 

A linear differential equation with constant coefficients is a differential equation of the form

a_n d^n y/dx^n + a_{n-1} d^{n-1} y/dx^{n-1} + ... + a_1 dy/dx + a_0 y = f(x)

where a_n, a_{n-1}, ..., a_1, a_0 are constants, and f(x) is a given function of x.

Such equations can be solved using the method of characteristic equations, which involves finding the roots of the characteristic equation, a_n r^n + a_{n-1} r^{n-1} + ... + a_1 r + a_0 = 0, and then using these roots to construct the general solution of the differential equation.

In some cases, the roots of the characteristic equation may be complex numbers, which leads to solutions in the form of complex exponentials. Linear differential equation with constant coefficients have a common solution that are called general solution.

Also in some cases the general solution can be represented with a linear combination of fundamental solutions( which are solution to homogeneous equation of the differential equation ) and a particular solution

A common example of a linear differential equation with constant coefficients is the second-order homogeneous equation

dy^2/dx^2 + 4 dy/dx + 4y = 0

The characteristic equation is r^2 + 4r + 4 = 0, which has roots r = -2 ± √(-4 - 4)i = -2 ± 2i. These are complex roots, so the general solution of the differential equation is of the form y(x) = e^(-2x) (c_1 cos(2x) + c_2 sin(2x)) Where c1 and c2 are arbitrary constant. This is a general solution of the differential equation.