A Linear Homogeneous Second-Order Differential Equation with Constant Coefficients
A Linear Homogeneous Second-Order Differential Equation with Constant Coefficients
This Demonstration shows how to solve a linear homogeneous differential equation with constant coefficients , where and are constant. First solve the characteristic equation +pr+q=0. If and are two real roots of the characteristic equation, then the general solution of the differential equation is , where and are arbitrary constants. If =, the general solution is x(c+dx). If , the general solution is (ccos(βx)+dsin(βx).
y''+py'+qy=0
p
q
2
r
r
1
r
2
cx+dx
r
1
e
r
2
e
c
d
r
1
r
2
r
1
e
r=α±βi
αx
e