Agenda for differential equations
1. Complex numbers
2. Differential calculus
3. Integral calculus
4. Modeling
5. Element equations
6. System equations
7. Differential equations
8. Solving differential equations
9. Differential equations
1
1. Complex numbers
Definition
Arithmetic
In-phase and quadrature
9. Differential equations
1. Complex numbers
2
Definition
A complex number, z, consists of the sum
of a real and imaginary number.
The symbols i and j have the value of the
square root of -1
Example
imaginary axis
a+bi
b
a=3
b=4
z = a + bi
z = 3 + 4i
r
real axis

a
9. Differential equations
1. Complex numbers
3
Arithmetic (1 of 2)
Addition: (a+bj) + (c+dj) = (a+c)+(b+d)j
Subtraction: (a+bj) - (c+dj) = (a-c)+(b-d)j
Multiplication: (a+bj)(c+dj) = (ac-bd)+(cb+da)j
Conjugate: conj(a+bj) = a-bj
Absolute: abs(a+bj) = sqrt(a2+b2)
Argument: arg(a+bj) = atan2(b,a)
Division: (a+bj)/(c+dj) = (a+bj) conj(c+dj)/
[abs(c+dj)]2
a+bj = r x ej
where r = abs(a+bj ) and  = arg(a+bj )
9. Differential equations
1. Complex numbers
4
Arithmetic (2 of 2)
Complex numbers
c1
add
ImSum
1+2i
subtract
ImSub
1+2i
multiply
ImProduct 1+2i
division
ImDiv
1+2i
conjugate ImConjugate1+2i
absolute ImAbs
1+2i
argument ImArgument 1+2i
c2
3+4i
3+4i
3+4i
3+4i
results
4+6i
-2-2i
-5+10i
0.44+8E-002i
1-2i
2.24
1.11 63.4
Complex arithmetic using Excel
9. Differential equations
1. Complex numbers
5
In-phase and quadrature (I&Q)
In-phase = component of signal that
is in-phase with reference
Quadrature = component of signal
that is 90 degrees out of phase with
reference
9. Differential equations
1. Complex numbers
6
2. Differential calculus
Derivative of a function
Elementary derivative operations
Examples
Critical points
Partial differentiation
9. Differential equations
2. Differential calculus
7
Derivative of a function
Lim f (x)
x 0
9. Differential equations
f(x)
x
2. Differential calculus
8
Elementary derivative operations
D k = 0
D xn = nxn-1
D ln x = 1/x
D eax = a eax
9. Differential equations
2. Differential calculus
9
Examples (1 of 2)
D k f(x) = k D f(x)
D (f(x)  g(x)) = D f(x)  D g(x)
D (f(x) g(x)) = f(x) D g(x) + g(x) D f(x)
D (f(x)/g(x)) = [g(x) D f(x) - f(x) D g(x)]/g(x)2
D [f(x)]n = n[f(x)]n-1 D f(x)
D f (g(x)) = Dg (f(g)) Dx g(x)
9. Differential equations
2. Differential calculus
10
Examples (2 of 2)
D sin x = cos x
D cos x = -sinx
D tan x = sec2x
D arcsin x = 1/sqrt(1 - x2)
D arctan x = 1/(1 + x2)
9. Differential equations
2. Differential calculus
11
Critical points
f ‘ (x) = 0 at critical point
f “ (x) < 0 at maximum point
f “ (x) > 0 at minimum point
f “ (x) = 0 at inflection point
f (x)
local
maximum
inflection
point
local
minimum
global
minimum
singular
point
x
9. Differential equations
2. Differential calculus
12
Partial differentiation
A partial derivative is a derivative that is
taken with respect to only one variable
z = 4x3 - 5y2 + 2xy + y -12
z/ x = 12x2 + 2y
Partial derivatives are important in finite
element computations
9. Differential equations
2. Differential calculus
13
3. Integral calculus
Integration
Elementary integration operations
Examples
Integration by parts
Initial values
Definite integral
9. Differential equations
3. Integral calculus
14
Integration
Integration is the inverse operation of
differentiation
f ‘ (x) dx = f (x) + C
9. Differential equations
3. Integral calculus
15
Elementary integration operations
k dx = k x + C
xm dx = xm+1/(m+1) + C
e kx dx = ekx/k + C
9. Differential equations
3. Integral calculus
16
Examples
sin x dx = -cos x +C
1/x dx = | ln x | + C
ln x dx = x ln x - x + C
dx/(k2 + x2) = I/k arctan(x/k) + C
9. Differential equations
3. Integral calculus
17
Integration by parts (1 of 3)
Integration by parts is an integration
technique that is used when the function
can be partitioned into two parts with
favorable properties
f(x) dg(x) = f(x)g(x) -
9. Differential equations
3. Integral calculus
g(x) df(x) +C
18
Integration by parts (2 of 3)
f(x)
dg(x)
x2
ex dx
2x
ex
df(x)
g(x)
x2 ex dx = x2 ex -
9. Differential equations
ex (2x) dx + C
3. Integral calculus
19
Integration by parts (3 of 3)
f(x)
dg(x)
2x
ex dx
2
ex
df(x)
g(x)
ex (2x) dx = 2x ex -
ex (2) dx + C
= 2x ex - 2 ex
x2 ex dx = x2 ex - 2x ex + 2 ex + C
9. Differential equations
3. Integral calculus
20
Initial values
The constant of integration C can be
found only if the value of the function is
known at a point
If there are multiple integrations involved,
then multiple initial values are needed
Example, if f(x) = 4 when x = 1 then
(3x2 - 2x)dx = x3- x2 + C
13 - 12 + C = 4
C=4
9. Differential equations
3. Integral calculus
21
Definite integrals
A definite integral is restricted to the
region bounded by lower and upper limits
x2
f ‘(x) dx = f(x2 ) - f(x1)
x1
2
2x dx = x2(2) - x2(1) = 22 - 12 = 3
1
9. Differential equations
3. Integral calculus
22
4. Modeling
Approaches to finding a model
Linear systems
Nonlinear systems
Guidelines for equations
9. Differential equations
4. Modeling
23
Approaches to finding a model
1. Lumped parameters
• Break system into smaller elements
• For each element, use the physical laws
that govern the element to write
equations
• Build a model of the system from these
lumped parameters
2. System identification
• Stimulate the system and observe its
response
• Works only with existing systems
9. Differential equations
4. Modeling
24
Linear systems (1 of 3)
A system is linear if and only if it obeys
the principle of superposition
• H(x1 +  x2) = H(x1) + H(x2), where H
is the system response
9. Differential equations
4. Modeling
25
Linear systems (2 of 3)
system response
x1
H
y = H(x1 + x2)
x2
9. Differential equations
x
4. Modeling
26
Linear systems (3 of 3)
y1 +y2
slope K
y2
y1
x1
9. Differential equations
x2
4. Modeling
x1 +x2
27
Nonlinear systems (1 of 3)
Occasionally, application of physical laws
to a system result in nonlinear equations.
The nonlinearity may be overcome by
finding a limited region of operation where
linear operation takes place
9. Differential equations
4. Modeling
28
Nonlinear systems (2 of 3)
(y1 +y2)
slope K
y2
y1
c
x1
9. Differential equations
x2
x1 +x2
4. Modeling
29
Nonlinear systems (3 of 3)
(y1 +y2) y2
y1
c
x1
9. Differential equations
x2
4. Modeling
x1 +x2
30
Guidelines for equations (1 of 4)
1. Understand the system -- sketch or
describe in qualitative terms
2. Identify inputs and outputs, including
disturbances
3. Express system in terms of elements
that can be expressed mathematically
4. Develop equations for each element
9. Differential equations
4. Modeling
31
Guidelines for equations (2 of 4)
5. Determine unknown parameter values
by analysis or experiment
6. Adjust the model until it produces
behavior like the actual system
7. Simplify the system if nonlinearities are
involved
9. Differential equations
4. Modeling
32
Guidelines for equations (3 of 4)
Ideally, the relationship should be linear
A lumped-parameter model has time as its
only independent variable. This fact allows
ordinary differential equations to be used. If
there are more independent variables, partial
differential equations would need to be used,
and they are more difficult
Use idealized equivalent of the system; e.g.
• Mass concentrated at a point rather than
distributed
• Inductors have no resistance or
capacitance
9. Differential equations
4. Modeling
33
Guidelines for equations (4 of 4)
The number of variables and the number
of equations needs to be the same.
Units need to be consistent
Need to validate the model with
prototypes or data from similar systems
In practice, systems are not truly linear.
Variations in the plant or transducers can
make design much harder
9. Differential equations
4. Modeling
34
5. Element equations
Proportional (P) relationship
Integral (I) relationship
Derivative (D) relationship
PID
Electrical components
Rectilinear mechanical components
Rotational mechanical components
Fluid component
Thermal components
9. Differential equations
5. Element equations
35
Proportional (P) relationship
v(t)
i(t)
i(t)
a
b
R
i(t) = current (A) = through variable
v(t) = voltage (V) = across variable
R = resistance ()
i(t) = 1/R v(t)
through variable = constant * across variable
9. Differential equations
5. Element equations
36
Integral (I) relationship
v(t)
i(t)
i(t)
a
b
L
i(t) = current (A) = through variable
v(t) = voltage (V) = across variable
L = inductance (H)
i(t) = 1/L
v(t) dt
through variable = constant * ( across variable) dt
9. Differential equations
5. Element equations
37
Derivative (D) relationship
v(t)
i(t)
i(t)
a
b
C
i(t) = current (A) = through variable
v(t) = voltage (V) = across variable
C = capacitance (F)
i(t) = C d/dt v(t)
through variable = constant * d/dt( across variable)
9. Differential equations
5. Element equations
38
PID
Proportional (P) -- through variable is
proportional to across variable
Integral (I) -- through variable is
proportional to integral of across variable
Derivative (D) -- through variable is
proportional to derivative of across
variable
9. Differential equations
5. Element equations
39
Electrical components
•Across variable: potential difference v (V)
•Through variable: current I (A)
P -- Resistor
R()
I -- Inductor
L(H)
D -- Capacitor
C(F)
9. Differential equations
5. Element equations
40
Rectilinear mechanical components
•Across variable: linear velocity v(m/s)
•Through variable: force f(N)
P -- Linear damper
B(N/ms-1)
I -- Linear spring
K(N/m)
D -- Mass
M(kg)
9. Differential equations
5. Element equations
41
Rotational mechanical components
•Across variable: angular velocity (rad/s)
•Through variable: torque T(Nm)
P -- Angular damper
B(Nm/rads-1)
I -- Angular spring
K(Nm/rad)
D -- Inertia
J(Nm/rads-2)
9. Differential equations
5. Element equations
42
Fluid components
•Across variable: pressure head h(m)
•Through variable: volume flow rate q(m 3s-1)
P -- fluid resistance
1/R(m2/s)
D -- fluid capacity
A(m2)
9. Differential equations
5. Element equations
43
Thermal components
•Across variable: temperature difference (K)
•Through variable: heat flow rate q(W)
P -- thermal resistance
1/R(W/K)
D -- thermal capacity
C(J/K)
9. Differential equations
5. Element equations
44
6. System equations
Example -- suspension
9. Differential equations
6. System equations
45
Example -- suspension
body
displacement
x(t)
body mass
spring, k
shock absorber, b
m d2x/dt2 = -b dx/dt - k x
wheel
9. Differential equations
6. System equations
46
7. Differential equations (de)
Definition of de
Order of a de
Linear de
Linear de with constant coefficients
Nonlinear de
Homogeneous de
Nonhomongeneous de
Auxiliary equation
9. Differential equations
7. Differential equations
47
Definition of de
A differential equation is a mathematical
expression combining a function (e.g.,
y=f(x)) and one or more of its derivatives
Examples
• dy/dx - 5 y = 0
• d2y/dx2 - 3 dy/dx + 2y = 0
• d2y/dx2 - (x2+5) [dy/dx]2 + y = sin 2x
9. Differential equations
7. Differential equations
48
Order of a de
The order of a differential equation is the
order of the highest derivative in the
equation
Examples
• dy/dx - 5 y = 0 -- 1st
• d2y/dx2 - 3 dy/dx + 2y = 0 -- 2nd
• d2y/dx2 - (x2+5) [dy/dx]2 + y = sin 2x -- 2nd
9. Differential equations
7. Differential equations
49
Linear de
A linear differential equation is an equation
consisting of a sum of terms each made of a
multiplier and either the function or its
derivatives
Examples
• dy/dx - 5 y = 0 -- linear
• d2y/dx2 - 3 dy/dx + 2y = 0 -- linear
• d2y/dx2 - (x2+5) [dy/dx]2 + y = sin 2x -- nonlinear
9. Differential equations
7. Differential equations
50
Linear de with constant coefficients
If the multipliers are constant, then the
differential equation is said to have
constant coefficients
Examples
• dy/dx - 5 y = 0 -- constant coefficients
• dy/dx - 5 xy = 0 -- non- constant
9. Differential equations
7. Differential equations
51
Nonlinear de
If the function or one of its derivatives is
raised to a power or embedded in another
function, the differential equation is
nonlinear
Example
• d2y/dx2 - (x2+5) [dy/dx]2 + y = sin 2x -nonlinear
9. Differential equations
7. Differential equations
52
Homogeneous de
A homogeneous differential equation is
one in which each term contains either the
function or its derivatives. In other words,
the sum of the derivative terms is zero
Examples
• dy/dx - 5 y = 0 -- homogeneous
• d2y/dx2 - 3 dy/dx + 2y = 0 -homogeneous
9. Differential equations
7. Differential equations
53
Nonhomogeneous de
A nonhomogeneous differential equation
is a sum of derivative terms that doesn’t
equal zero
Example
• d2y/dx2 - (x2+5) [dy/dx]2 + y = sin 2x -non-homogeneous
9. Differential equations
7. Differential equations
54
Auxiliary equation
The auxiliary equation is the polynomial
formed by replacing all derivatives in a linear,
constant coefficient, homogeneous
differential equation with variables raised to
the the power of the respective derivatives
Example
• d2y/dx2 - 3 dy/dx + 2y = 0 has an auxiliary
equation of s2 - 3s + 2 = 0
9. Differential equations
7. Differential equations
55
8. Solving differential equations
Introduction
Examples
Alternate expression
9. Differential equations
8. Solving differential equations
56
Introduction
There are a large number of types of
differential equations
Many types have closed form solutions;
others do not
A type of differential equations of
importance to engineering is the linear,
non-homogeneous differential equation
with constant coefficients
9. Differential equations
8. Solving differential equations
57
Example 1
de: Dy - 2y = 0
auxiliary equation: S - 2 = 0
root: +2
solution: y = C e+2x
if y(0) = 10, then C = 10
9. Differential equations
8. Solving differential equations
58
Example 2
de: D2y + 3 Dy + 2y = 0
auxiliary equation: S2 + 3S + 2 = 0
roots: -1, -2
solution: y = C1 e-2x + C2 e-x
if y(0) = 0, Dy(0) = -1, then C1 =1 and C2 = -1
9. Differential equations
8. Solving differential equations
59
Example 3
de: D2y + y = 0
auxiliary equation: S2 + 1 = 0
roots: +i, -i
solution: y = C1 cos x + C2 sin x
9. Differential equations
8. Solving differential equations
60
Example 4
de: D2y +2Dy + 2y = 0
auxiliary equation: s2 + 2s + 2 = 0
roots: -1 + i, -1 - i
solution: y = C1 e-x cos x + C2 e-x sin x
9. Differential equations
8. Solving differential equations
61
Example 5
de: D2y +2Dy + y = 0
auxiliary equation: S2 + 2S +1 = 0
roots: -1 , -1
solution: y = (C1 + C2 x ) e-x
9. Differential equations
8. Solving differential equations
62
Example 6
de: D5y = 0
auxiliary equation: S5 = 0
roots: 0, 0, 0, 0, 0
solution: y = C1 + C2 x + C3x2 + C4 x3 + C5x4
9. Differential equations
8. Solving differential equations
63
Example 7
de: D4y + 4 D3y +8 D2y + 8 Dy +4 y = 0
auxiliary equation: s4 + 4 s3 +8 s2 + 8 s +4
= (s2 + 2s + 2)( s2 + 2s + 2) = 0
roots: -1 + i, -1 - i, -1 + i, -1 - i
solution: y = (C1 + C2 x) e-x cos x + (C3 +
C4 x) e-x sin x
9. Differential equations
8. Solving differential equations
64
Example 8 (1 of 2)
de: D2y + Dy - 2y = 2x -40 cos 2x
homogeneous auxiliary equation: s2 + s - 2 = 0
homogeneous roots: 1, -2
homogeneous solution: yc = C1 e+x + C2 e-2x
particular roots: 0, 0, +2i, -2i
particular solution: yp = A + Bx + C cos 2x + E
sin 2x
total solution: y = yc + yp
9. Differential equations
8. Solving differential equations
65
Example 8 (2 of 2)
-2 yp = -2A -2Bx -2C cos 2x -2E sin 2x
D yp = B + 2Ecos2x - 2C sin 2x
D2 yp =-4C cos 2x -4E sin 2x
constant terms : -2A + B =0
X terms: -2B = 2
cos x terms: -2C + 2E -4C = -40
sin x terms: -2E -2C -4E = 0
constants: A = -0.5. B = -1, C = 6, E = -2
9. Differential equations
8. Solving differential equations
66
Example 9 (1 of 2)
de: D2y + y = sin x
homogeneous auxiliary equation: s2 + 1 = 0
homogeneous roots: +i, -i
homogeneous solution: yc = C1 cos x + C2 sin x
particular roots: +i, -i
particular solution: yp = Ax cos x + Bx sin x
total solution: y = yc + yp
9. Differential equations
8. Solving differential equations
67
Example 9 (2 of 2)
yp = Ax cos x + Bx sin x
D yp = A cos x - Ax sin x + B sin x +Bx cos x
D2 yp = -2A sin x - Ax cos x + 2B cos x - Bx
sin x
cos x terms: 2B = 0
sin x terms: -2A = 1
constants: A = -0.5, B = 0
9. Differential equations
8. Solving differential equations
68
Example 10 (1 of 1)
de: D3y - Dy = 4 e-x + 3 e2x
homogeneous auxiliary equation: s3 - s = 0
homogeneous roots: 0, +1, -1
homogeneous solution: yc = C1 + C2 e+x + C3 e-x
particular roots: -1, 2
particular solution: yp = Ax e-x + B e2x
total solution: y = yc + yp
9. Differential equations
8. Solving differential equations
69
Example 10 (2 of 2)
yp = Ax e-x + B e2x
D yp = A e-x - Ax e-x + 2 B e2x
D2 yp =-2A e-x + Ax e-x + 4 B e2x
D3 yp =3A e-x - Ax e-x + 8 B e2x
e-x terms: -A + 3A = 4
e2x terms: -2B + 8B = 3
constants: A = 2. B = 0.5
9. Differential equations
8. Solving differential equations
70
Example 11
In the previous problem, y(0) = 0, Dy(0) =
-1, D2 y(0) = 2
Determine C1, C2, C3
Use the general solution: y = C1 + C2 e+x +
C3 e-x + 2x e-x + 0.5 e2x
Dy = C2 e+x - C3 e-x - 2x e-x + 2e-x + e2x
D2 y = C2 e+x + C3 e-x + 2x e-x - 4e-x + 2e2x
y(0) = 0 = C1 + C2 + C3 + 0.5
Dy(0) = -1 = C2 - C3 + 3
D2 y(0) = 2 = C2 + C3 -2
C1 = -4.5, C2 = 0, C3 = 4
9. Differential equations
8. Solving differential equations
71
Example 12 (1 of 3)
de: D2 y + 2D y + 2y = cos x
homogeneous auxiliary equation: s2 + 2s +
2 =0
homogeneous roots: -1+i, -1-i
homogeneous solution: yc = C1 e-x cos x +
C2 e-x sin x
particular roots: +i, -i
particular solution: yp = A cos x + B sin x
total solution: y = yc + yp
9. Differential equations
8. Solving differential equations
72
Example 12 (2 of 3)
yp = A cos x + B sin x
D yp = - A sin x + B cos x
D2 yp = - A cos x - B sin x
cos x terms: -A +2B +2A = 1
sin x terms: -B -2A + 2B = 0
constants: A = 0.2, B = 0.4
9. Differential equations
8. Solving differential equations
73
Example 12 (3 of 3)
Use the general solution: y = C1 e-x cos x + C2
e-x sin x + 0.2 cos x + 0.4 sin x
initial conditions: y(0) = 1, D y(0) = 0
Dy = - C1 e-x cos x - C2 e-x sin x - C1 e-x sin x +
C2 e-x cos x - 0.2 sin x + 0.4 cos x
y(0) = 1 = C1 + 0.2
Dy(0) = 0 = - C1 + C2 + 0.4
C1 = 0.8, C2 = 0.4
y(x) = 0.8 e-x cos x + 0.4 e-x sin x + 0.2 cos x +
0.4 sin x
9. Differential equations
8. Solving differential equations
74
Alternate expression (1 of 3)
It is sometimes desirable to express a
higher-order differential equation as a set
of first-order equations
• Matrix representation
• Computer solutions
9. Differential equations
8. Solving differential equations
75
Alternate expression (2 of 3)
Example
• D3y + 2 D2Y + 5Dy + 10y = r
• Choose
• y1 = y
• y2 = Dy = Dy1
• y3 = D2y = Dy2
• Single equation replaced by three
equations
• Dy1 = y2
• Dy2 = y3
• Dy3 = r - 10 y1 - 5y2 - 2y3
9. Differential equations
8. Solving differential equations
76
Alternate expression (3 of 3)
• Matrix format
Dy1
Dy2
Dy3
9. Differential equations
=
0 1 0
0 0 1
-10 -5 -2
y1
y2
y3
8. Solving differential equations
+
0
0
r
77