1. Homogeneous Equations with Constant Coefficients

A second order ordinary differential equation has the form

(((d^2)y)/(d (x^2)))=f(x,y,((d y)/(d x))).

where f is some given function.  The DE is linear if it has a form

y''+p(x)y'+q(x)y=g(x).

If the equation can not be written in this form, then it is nonlinear.  The DE is (linear and) constant coefficients, if

a y''+b y'+c y=g(x),

where a, b, and c are constants.  It is homogeneous if g(x)=0.

Fundamental solutions to Second Order, Constant Coefficient, Homogeneous Equations

Looking at the equation a y''+b y'+c y=0, assume a solution of the form y=(e^(r x)).  Plug into the DE to get

a (r^2)(e^(r x))+b r (e^(r x))+c (e^(r x))=0  ⟹  a (r^2)+b r+c=0.

For the DE a y''+b y'+c y=0, the characteristic equation is a (r^2)+b r+c=0.  If the characteristic equation has real, distinct zeros, (Subscript[r, 1]) and (Subscript[r, 2]), then the fundamental solutions for the DE are

(Subscript[y, 1])=(e^((Subscript[r, 1])x)) and (Subscript[y, 2])=(e^((Subscript[r, 2])x)).

Superposition

Theorem:  If a linear, homogeneous ODE has fundamental solution (Subscript[y, 1])(x) and (Subscript[y, 2])(x), then the general solution to the ODE is given by y(x)=(Subscript[c, 1])(Subscript[y, 1])(x)+(Subscript[c, 2])(Subscript[y, 2])(x), where (Subscript[c, 1]) and (Subscript[c, 2]) are arbitrary constants.

Examples

Example 1:  y''+y'-6y=0.

The characteristic equation is (r^2)+r-6=0.  This factors as (r+3)(r-2)=0, so r=-3 or r=2.  Thus, the fundamental solutions are (Subscript[y, 1])=(e^(-3x)) and (Subscript[y, 2])=(e^(2x)).  The general (family of) solutions is given by y=(Subscript[c, 1])(e^(-3x))+(Subscript[c, 2])(e^(2x)).

Example 2:  y''+3y'+2y=0.

Example 3:  y''-7y'+12y=0.