Method of Variation of Parameters for Second-Order Linear Differential Equations with Constant Coefficients
Method of Variation of Parameters for Second-Order Linear Differential Equations with Constant Coefficients
This Demonstration shows how to solve a nonhomogeneous linear second-order differential equation of the form , where and are constants.
y''+py'+qy=f(x)
p
q
The corresponding homogeneous equation is with the characteristic equation +pr+q=0. If and are two real roots of the characteristic equation, then the general solution of the homogeneous 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
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
To find a particular solution of the nonhomogeneous equation, the method of variation of parameters (Lagrange's method) is used. The solution has the form , where and are independent partial solutions of the corresponding homogeneous equation, and and are functions of satisfying the system of equations
y=A+B
y
1
y
2
y
1
y
2
A
B
x
A'+B'=0
y
1
y
2
A''+B''=f(x).
y
1
y
2
Define the Wronskian by . Then , .
w='-'
y
1
y
2
y
2
y
1
A'=-f(x)/w
y
2
B'=f(x)/w
y
1
The general solution of the nonhomogeneous equation is the sum of the general solution of the homogeneous equation and a particular solution of the nonhomogeneous equation.