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Warm–up Problems
1) Find the general solution of
y – 5y + 6y = 8 + 2sin x.
2) How would your work change if the
equation was y + y = 8 + 2sin x?
Variation of Parameters –– Another
Method for Finding yp
1) Find yc = c1y1 + c2y2
2) Find the Wronskian, W, of y1 and y2.
3) Put the DE in the form y + Py + Qy = f (x)
0
y2
y1
0
4) Find W1 
and W2 
f  x  y2
y1 f  x 
W1
W2


dx
5) Find u1   dx and u2  
W
W
6) yp = u1y1 + u2y2
Ex. Solve y – 4y + 4y = (x + 1)e2x
Ex. Solve 4y + 36y = csc 3x
For higher order:
y1
y2
y3
W  y1
y2
y3
y1
y2
y3
y1
W2  y1
y1
W1 
0
y2
y3
0
y2
y3
f  x
y2
y3
0
y3
y1
y2
0
0
y3
W3  y1
y2
0
f  x
y3
y1
y2
f  x
Ex. Solve y + 4y = sec 2x
Practice Problems
Section 4.6
Problem 2