A Linear Homogeneous Second-Order Differential Equation with Constant Coefficients

p

2

q

1

characteristic equation

solution of characteristic equation

particular solutions

general solution

Solve the differential equation:

′′

y

(x)+2

′

y

(x)+y(x)0

2

r

+2r+10

This Demonstration shows how to solve a linear homogeneous differential equation with constant coefficients

y''+py'+qy=0

, where

p

and

q

are constant. First solve the characteristic equation

2

r

+pr+q=0

. If

r

1

and

r

2

are two real roots of the characteristic equation, then the general solution of the differential equation is

c

r

1

x

e

+d

r

2

x

e

, where

c

and

d

are arbitrary constants. If

r

1

=

r

2

, the general solution is

r

1

x

e

(c+dx)

. If

r=α±βi

, the general solution is

αx

e

(ccos(βx)+dsin(βx)

.