Part I Problems and Solutions
In each of the following three problems find a particular solution to the differential equation. Use complex exponentials where possible.
Problem 1: y(3) + y00 − y0 + 2y = 2 cos x
Solution: characteristic polynomial p(s) = s3 + s2 − s + 2; complex replacement: z(3) +
z00 − 2z0 + 2z = 2eix . Complex solution from ERF:
zp =
2eix
2eix
2(1 + 2i )ei x
= 3
=
p (i )
i + i2 − i + 2
(1 − 2i )(1 + 2i )
Thus
zp =
2 + 4i
(cos x + i sin x )
5
, and
y p = Re(z p ) =
2
4
cos x − sin x
5
5
Problem 2: y00 − 2y0 + 4y = e x cos x
Solution: complex replacement: z00 − 2z0 + 4z = e(1+i)x . p(s) = s2 − 2s + 4; p(1 + i ) =
(1+ i ) x
(1 + i )2 − 2(1 + i ) + 4 = 2, so z p = e 2 , and thus
y p = Re(z p ) =
1 x
e cos x
2
Problem 3: y00 − 6y0 + 9y = e3x
Solution: p(s) = s2 − 6s + 9 = (s − 3)2 so y p = cx2 e3x . p(3) = p0 (3) = 0, p00 (3) = 2 6= 0.
This the generalized ERF gives
1
y p = x2 e3x
2
Problem 4: Find the real general solution to the DE
d3 x
− x = e 2t
dt3
Part I Problems and Solutions
OCW 18.03SC
Solution: p(r ) = r3 − 1, p(2) = 23 − 1 = 7 6= 0. Thus the ERF gives a particular solution
to the inhomogenous DE x (3) − x = e2t as x p = p(12) e2t = 17 e2t .
xh solution to the homogeneous eqn.x (3) − x = 0, characteristic equation p(r ) = r3 − 1 =
0 → r3 = 1 → cube roots of 1. √
√
r = 1, e2πi/3 , e4πi/3 = 1, − 12 + i 23 , − 12 − 23
√ √ So x1 = et , x2 = e−t/2 cos 23 t , x3 = e−t/2 sin 23 t
Real solutions: xh = c1 x1 + c2 x2 + c3 x3 so the general solution x = x p + xh is
√ !
√ !
1 2t
3
3
t + c3 e−t/2 sin
x = e + c1 et + c2 e−t/2 cos
t
7
2
2
Problem 5: Find a particular solution to the differential equation
y00 − 4y =
1 2x
e + e−2x
2
Solution: y p = y p,1 + y p,2 where p( D )y p,1 = 12 e2x and p( D )y p,2 = 12 e−2x by superposition
principle. p(r ) = r2 − 4 so p(2) = p(−2) = 0. Using the generalized ERF p0 (r ) = 2r. Thus,
y p,1 =
y p,2 =
1
2
p 0 (2)
1
2
p0 (−2)
xe2x =
1
xe−2x = − xe−2x
8
y p = y p,1 + y p,2 =
2
1 2x
xe
8
x 2x
e − e−2x
8
MIT OpenCourseWare
http://ocw.mit.edu
18.03SC Differential Equations
Fall 2011
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