Transcript Section 4.7 - Gordon State College

Section 4.7
Variation of Parameters
METHOD OF VARIATION OF
PARAMETERS
For a second-order linear equation in standard form
y″ + Py′ + Qy = g(x).
1. Find the complementary function yc(x) = c1y1(x) + c2y2(x).
2. Replace the constants by the functions u1 and u2 to form
the particular solution yp(x) = u1(x)y1(x) + u2(x)y2(x).
3. Solve the following system of equations for u′1 and u′2
u1 y1  u2 y2  0
u1 y1  u2 y2  g ( x)
4. Integrate to find u1(x) and u2(x).
USING CRAMER’S RULE TO
SOLVE THE SYSTEM
The system of equations
u1 y1  u2 y2  0
u1 y1  u2 y2  g ( x)
can be solved using Cramer’s Rule (determinants).
W1
W2
u1 
and u2 
where
W
W
y1 y2
0
y2
y1
W
, W1 
, and W2 
y1 y2
g ( x) y2
y1
0
g ( x)
COMMENTS ON VARIATION OF
PARAMETERS
1. The system of equations can be solved by other
methods as well. That is, by substitution or
elimination.
2. The method of Variation of Parameters is not
limited to g(x) being either a polynomial,
exponential, sine, cosine, or finite sums and
products of these functions.
HIGHER-ORDER EQUATIONS
Variation of Parameters can be used to solve higher-order
equations. Let
yp = u1(x)y1(x) + u2(x)y2(x) + . . . + un(x)yn(x)
Solve the following system of equations.
y1u1 
y1u1 
y2u2   
y2 u2   
ynun  0
yn un  0

y 1( n 1)u1  y (2n 1)u2    y (nn 1)un  f ( x)
HOMEWORK
1–33 odd