MTH 322–Differential Equations
Homework Problems Spring 2013 (Subject to Change)
Section numbers are from 7th Edition of text
Due dates will be announced in class
You must show your work for full credit
Section
Calculus
Refresher #1
Calculus
Refresher #2
9.1
Algebra
Refresher
9.2
Problems
Section
Handout
17.1
Handout
17.2(a)
1, 2, 3, 5, 7, 14, 16
1–5, 9, 10, 12
17.2(b)
20, 21, 23
Handout
17.3
1, 3, 5, 13, 15
1, 3–6, 14, 21, 22
17.4
1, 3, 5
E&PL
Handout
9.3(a)
2, 4, 5, 6, 10, 12, 16
9.3(b)
29, 31, 45, 47
9.5(a)
1–4, 5, 7, 9, 15, 17
9.5(b)
Partial
Derivatives
Exact
Problems
2, 4, 6, 10, 11, 18, 20, 23,
27, 28, 29
Matrices
Handout
Systems
Partial Fraction
Decomposition
Handout
Laplace
Handout
Inverse Laplace
Handout
Handout
Solving IVPs
Handout
Handout
9.4
1, 4, 5
9.6
2, 3, 6, 8
27, 33
Handout
MTH 322–Differential Equations Homework Handout
Calculus Refresher #1
Instructions: Find the first derivative.
(1)
(2)
(3)
(4)
y = −13ex
f (ψ) = cos ψ
y = sin x
y = tan θ
Instructions:
Z
(11)
Z
(12)
Z
(13)
Z
(14)
(5)
(6)
(7)
(8)
g(φ) = sec φ
y = e4x
h(r) = r3 e−5r
y = ln(3x5 + 2ex + 19)
(9) g(x) = 5 sin πx
(10) y = tan x sin x
Evaluate the integrals.
Z
x
e dx
(15)
e
cos r dr
(19)
(16)
Z
dr
Z
tan ψ sec ψ dψ
(17)
(20)
1 − y2
dy
z −1 dz
sec2 y dy
Z
sin θ dθ
1
p
Z
1
dw
w
−r
Z
(18)
x2
1
dx
+1
Calculus Refresher #2
Instructions: Evaluate the integrals.
Z
8
dx
(1)
5x + 3
Z
11
(2)
dw
7w − 2
Z
41
dr
(3)
31r + 26
Z Z
a
(4)
dx
bx + c
Z
(5)
10x
ke
Z
dx
(9)
Z
(6)
Z
cos 2θ dθ
Z
(7)
Z
(8)
(10)
3
3y 2 ey dy
xex dx
1
sin x dx
3
er sin er dr
Algebra Refresher
Instructions: Simplify 1–4; solve for y in 5–10.
(1) y = eln x
(5)
ln y = 2x3 + 7
(2) y = e7 ln x
(6)
− ln x
(7)
ln y = ln 9x + C
√
ln y = 3 x + C
x4
ln y =
+C
4
(3) y = e
1
(4) y = e 3 ln(x
4 +22)
(8)
(9)
(10)
x
+C
32
tan−1 y = 8x5 + C
sin−1 y =
Equilibria and the Phase Line (E&PL)
Instructions: For each of the following differential equations,
(a) Sketch the phase lines for the given differential equation.
(b) Identify each of the equilibrium points as a sink, source, or node.
(1)
dy
= y 2 − 10y + 16
dx
(4)
dx
= x3 + 7x2 − 18x
dt
(2)
dy
= 24 + 5y − y 2
dt
(5)
dP
=
dt
(3)
dQ
= Q2 − 14Q + 51
dt
(6)
dw
= 3(w2 − 4w)2 (2w − 4)
dt
P
1−
15
P
− 1 P3
10
Partial Derivatives
Instructions: Find the specified partial derivative.
(1) f (x, y) = 3y 7 − 5x2 y 4 ; fx , fy
(2) g(r, θ) = 10r3 sin 4θ; gr , gθ
(3) R(p, q) = ln(3p5 + 5p2 q 4 − 10q 8 ); Rp , Rq
x
(4) w = , wx , wy
y
√
(5) h(x, y) = 3e2x−4y ; hx , hy
(6) Q = x2 ln(3y 2 − 7y); Qx , Qy
(7) P = 8r6 − 7t; Pr , Pt
(8) y = tan(3θ) sec(r4 ); yr , yθ
Exact Equations
Instructions: Determine whether the given equation is exact. If exact, solve.
1
2
2
−xy
(1) (2xy − 3)dx + (2x y + 4)dy = 0
(3) (y ln y − e )dx +
+ x ln y dy = 0
y
(2)
3
3
1 − + y dx + 1 − + x dy = 0
x
y
(4) x
dy
= 2xex − y + 6x2
dx
Instructions: Solve the initial value problem (IVP).
(5) (4y + 2x − 5)dx + (6y + 4x − 1)dy = 0, y(−1) = 2
Matrices
Instructions: Calculate the determinant. Find the eigenvalue(s) and eigenvector(s) if they exist.
2 3
2 4
(1) A =
(4) A =
2 1
−1 6
1 2
1 3
(2) B =
(5) B =
− 12 1
3 1
−1 3
1/2
0
(3) C =
(6) C =
−3 5
1 −1/2
Systems of Equations
Instructions: Solve the system. If an initial condition is given, solve the IVP.
2 3
2 4
−1
0
0
(1) x =
(4) x =
, x(0) =
2 1
−1 6
6
1 2
1 3
3
(2) x0 =
(5) x0 =
, x(0) =
− 12 1
3 1
1
−1 3
1/2
0
3
0
0
(3) x =
(6) x =
, x(0) =
−3 5
1 −1/2
5
Partial Fraction Decomposition
Instructions: Find the partial fraction decomposition for the rational functions.
7x + 10
2
x + 3x + 2
17x − 67
(2) g(x) = 2
x − 8x + 15
(1) f (x) =
12x2 + 5x + 2
(3x2 + 3)(x − 2)
45x2 − x + 44
(4) f (x) =
(x + 1)(5x2 + 5)
(3) h(x) =
(5) g(x) =
Laplace Transform
1. Compute L{t} using the definition, NOT the table.
2. Find the Laplace transforms of the given functions using the table.
(a) f (t) = 15
(b) g(t) = 7t5 + cos 3t
(c) h(t) = 10 sin(πt) − 24 sinh(et)
(d) f (t) = 4t cos(3t) + t9/2
(e) g(t) = 3 sin(qt) + qt cos(qt)
(f) h(t) = −9e5t + 13e4t cos(7t)
9x2 + 4x + 9
x3 + x2
Inverse Laplace
Instructions: Find the inverse transform of each of the following.
(1) F (s) =
(2) G(s) =
(3) H(s) =
(4) F (s) =
(5) G(s) =
(6) H(s) =
7
23
s2 − 81
+
− 2
s s − 5 (s + 81)2
s2
11s
27
− 2
+ 81 s + 81
14
37s
− 2
2
(s + 4) − 49 s − 19
8s
18
10
+ 2
−
+ 55 s − 36 7s − 4
5s2
s2
s+3
−s−2
1
(s + 1)(s2 + 1)
Solving IVPs. . .
Instructions: Solve the following initial value problems using Laplace transforms.
(1) y 0 − 5y = 0;
y(0) = 2
(2) y 0 − 5y = e5t ;
y(0) = 0
(3) y 0 + y = sin t;
y(0) = 1
(4) y 00 + 4y = 0;
y(0) = 2, y 0 (0) = 2
(5) y 00 − 4y 0 + 4y = 0;
(6) y 00 − 9y 0 + 18y = 54;
y(0) = 0, y 0 (0) = 3
y(0) = 0, y 0 (0) = −3
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