hw25
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moienMath 21b: Linear Algebra Spring 2016
Homework 25: Differential equations II
This homework is due on Monday, April 11, respectively on Tuesday, April 12, 2016.
2
1 a) Find the solution to the differential equation ddtx2 = −x with ini-
dx
tial conditions
x(0) = 5, dt (0) = 0 by writing the initial condition
5
~v = as a linear combination of eigenvectors of A, where
0
0
x (t) x(t)
= A
0
y (t) y(t)
is the system written in vector form using 2 × 2 matrix A.
dx
2 Match the
differential
equations
dt
= Ax with
the
phase portraits.
0 1 0 −1 1 0
i) A = , ii) A = iii) A =
1 0 1 0 0 1
0 1 0 1 0 0
iv) A = v) A = vi) A =
−1 1/2 −1 −1/2 1 0
1.0 1.0 1.0
0.5 0.5 0.5
0.0 0.0 0.0
-0.5 -0.5 -0.5
-1.0 -1.0 -1.0
a) -1.0 -0.5 0.0 0.5 1.0 b) -1.0 -0.5 0.0 0.5 1.0 c) -1.0 -0.5 0.0 0.5 1.0
1.0 1.0 1.0
0.5 0.5 0.5
0.0 0.0 0.0
-0.5 -0.5 -0.5
-1.0 -1.0 -1.0
d) -1.0 -0.5 0.0 0.5 1.0 e) -1.0 -0.5 0.0 0.5 1.0 f) -1.0 -0.5 0.0 0.5 1.0
3 Determine the stability of the systems
1
1 0 1
−2 0 0 0
0 −1 1 0 1 −1 0 0
a)v 0(t) = v(t), b)v 0(t) = v(t) .
−1 −1 1 0 −1 −1 −3 −4
1 −1 1 −1 1 −1 4 −3
4 To solve the fourth order equation w0000(t) = w(t), we write it as
a system of first order differential equations of the form ~v 0(t) =
A~v (t) using ~v (t) = (x(t), y(t), z(t), w(t)) where w0(t) = z(t), z 0(t) =
y(t), y 0(t) = x(t), x0(t) = w(t) and A is a 4 × 4 matrix. Write
down a general closed form solution formula ~v (t) = c1eλ1t~v1(t) +
· · · + c4eλ4t~v4(t), where c1, c2, c3, c4 are parameters. Is this system
stable or unstable?
5 True or false? Give short explanations
a) If dx dx T
dt = Ax is stable, then dt = A x is stable.
−1
b) If dx dx
dt = Ax is stable, then dt = A x is stable.
c) If dx dx
dt = Ax is stable, then dt = −Ax is stable.
d) If dx dx
dt = Ax is stable, then dt = (A − In )x is stable.
Differential Equations II
d2 x
dt2 = −k 2x(t) is called harmonic oscillator. It has solu-
tions x(t) = a cos(kt) + b sin(kt), where a, b depend on initial
dy
conditions. It becomes a system dx 2
dt = y(t), dt = −k x(t). In
general, for a n × n matrix A, the system v 0 = Av is stable if
all eigenvalues satisfy Re(λj ) < 0.