CHAPTER 4. HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS 4.5. The Method of Variation of Parameters
In this section we consider a general method of determining a particular solution of the equation
a 0 (x) d 2 y
dx 2 + a 1 (x) dy
dx + a 2 (x)y = f (x); (1) where the functions a 0 (x); a 1 (x) and a 2 (x) are continuous on the interval [a; b].
Suppose that y 1 and y 2 are linearly independent solutions of the corresponding homogeneous equation
a 0 (x) d 2 y
dx 2 + a 1 (x) dy
dx + a 2 (x)y = 0: (2)
Then the complementary function of equation (2) is y c = c 1 y 1 (x) + c 2 y 2 (x)
The procedure in the method of variation of parameters is to replace the ar- bitrary constants c 1 and c 2 by the functions v 1 (x) and v 2 (x) which will be determined so that the resulting function
y p = v 1 (x)y 1 (x) + v 2 (x)y 2 (x) (3) will be a particular solution of equation (1): It can be seen that to determine v 1 (x) and v 2 (x); we have the following system of equations
v 1 0 (x)y 1 (x) + v 0 2 (x)y 2 (x) = 0 (4) v 1 0 (x)y 0 1 (x) + v 0 2 (x)y 2 0 (x) = f (x)
a 0 (x) for unknown functions v 1 0 and v 2 0 :
Since y 1 and y 2 are linearly independent solutions of equation (2); the determi- nant of coe¢ cients of this system
W (y 1 (x); y 1 (x)) = y 1 (x) y 2 (x) y 1 0 (x) y 2 0 (x)
= y 1 y 0 2 y 2 y 1 0 6= 0:
By Cramer’s Rule, the solution of system (4) is obtained as
v 0 1 (x) = 1 W (y 1 (x); y 1 (x))
0 y 2 (x)
f (x)
a
0(x) y 0 2 (x)
1
v 0 2 (x) = 1 W (y 1 (x); y 1 (x))
y 1 (x) 0 y 0 1 (x) a f (x)
0