R·I·T School of Mathematical Sciences
Homework 7
MATH 211 - 01
Due Monday October 27, 2014
1. Solve the following second order homogeneous differential equation.
4y 00 − 8y 0 − 5y = 0
4r2 − 8r − 5 = 0
(2r + 1)(2r − 5) = 0
r1 = −
1
2
r2 =
5
2
y = c1 e−t/2 + c2 e5t/2
2. Solve the following second order homogeneous differential equation.
y 00 + 14y 0 + 49y = 0
r2 + 14r + 49 = 0
(r + 7)(r + 7) = 0
r1 = r2 = −7
y = c1 e−7t + c2 te−7t
3. Solve the following second order homogeneous differential equation.
y 00 + 3y 0 + y = 0
r2 + 3r + 1 = 0
p
√
√
−3 ± 32 − 4(1)(1)
−3 ± 9 − 4
−3 ± 5
=
=
r=
2(1)
2
2
√
3
5
r=− ±
2
2
√
3
5
α=−
β=
2
2
"
√ !#
√ !
5
5
−3t/2
t + c2 cos
t
y=e
c1 sin
2
2
4. Solve the following initial value problem.
y 00 + 5y 0 + 6y = 0
y(0) = 0
r2 + 5r + 6 = 0
(r + 3)(r + 2) = 0
r1 = −3
r2 = −2
y = c1 e−3t + c2 e−2t
0 = c1 e0 + c2 e0
0 = c1 + c2
c1 = −c2
y = −c2 e−3t + c2 e−2t
y 0 = 3c2 e−3t − 2c2 e−2t
1 = 3c2 e0 − 2c2 e0
1 = c2
c1 = −1
y = −e−3t + e−2t
y 0 (0) = 1
5. Find a differential equation whose general solution takes the following
form.
y = c1 e3t + c2 et + c3 tet + c4 sin t + c5 cos t
r1 = 3
r2 = 1
r3 = r2 = 1
r4 = ±i
2
(r − 3)(r − 1)(r − 1)(r + 1) = 0
(r − 3)(r2 − 2r + 1)(r2 + 1) = 0
r5 − 5r4 + 8r3 − 8r2 + 7r − 3 = 0
y (5) − 5y (4) + 8y 000 − 8y 00 + 7y 0 − 3y = 0
6. Solve the following initial value problem.
y 00 + 6y 0 + 9y = 0
y(0) = 0
r2 + 6r + 9 = 0
(r + 3)(r + 3) = 0
r1 = r2 = −3
y = c1 e−3t + c2 te−3t
0 = c1 e 0 + 0
c1 = 0
y = c2 te−3t
1 = c2 e−3
c2 = e 3
y = e3 te−3t
y = te3−3t
y(1) = 1
7. Solve the following initial value problem.
y 00 + 16y = 0
y(0) = 0
r2 + 16 = 0
r2 = −16
r = ±4i
y = c1 sin (4t) + c2 cos (4t)
0 = 0 + c2
c2 = 0
y = c1 sin (4t)
1 = c1 sin (π/2)
c1 = 1
y = sin (4t)
y(π/8) = 1