Ordinary Differential Equations

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Ordinary Differential Equations
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What are derivatives? What are differential
equations?
Derivatives
Derivatives are gradients defined at particular
values of a function. In this case, y is a
differential function of x. They are also rates
of change of a quantity against another quantity.
It measures how quickly y changes with x.
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For nth order derivatives, we have
Differentiating n times !
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A differential equation (DE) is therefore an
equation that contains one or more derivatives of
a differentiable function.
Example
DE of order 3
The order of a DE is the order of the equations
highest order derivative. A DE is linear if it
can be written in the form y(n)(x)
pn-1(x)y(n-1)(x) p1(x)y(1) (x)
p0(x)y(x) r(x) (1) where pi, 0 i n -
1, and r are functions of x and y is
unknown. Note that y(n) has a coefficient of 1.
This is the standard form. If you have the
form f(x)y(n) , then divide throughout by
f(x) to get the standard form.
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An equation that cannot be written in the form
(1) is called nonlinear.
Examples of linear and nonlinear DE
First order linear DE
Second order linear DE
Second order nonlinear DE
If r(x) is identically zero, then (1)
becomes y(n)(x) pn-1(x)y(n-1)(x)
p1(x)y(1) (x) p0(x)y(x) 0 (2) and is
called homogeneous. If r(x) is not identically
zero, the equation is called non-homogeneous.
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Where and how do you get differential
equations? What does it mean to have a
differential equation?
Differential equations are what govern the laws
of Physics!
What are the important laws of Physics?
Laws of conservation of energy and momentum are
some of the fundamental laws of Physics. Example
Rate of change of energy flowing into a system
rate of change of energy flowing out of a
system rate of accumulation of energy inside
the system. These laws are used frequently to
determine the behaviour of nature around us.
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R is the resistance in the circuit which
represents loss. C is a capacitor which
represents storage. I represents the current
flow or the energy flow. Vi represents the
energy source or energy input.
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Laws of Physics that governs this circuit?
Let voltage across the capacitor be Vo Volts
Power input into the circuit IVi
Watts Power lost in the R I2R Watts Power
stored in the capacitor IVo Watts
By conservation of energy
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But, in a capacitor, the I and V relationship is
given by
Substituting into the previous equation
We have thus obtained a first order DE in a
simple circuit by considering simple laws of
Physics. C and R are constants. Vi may not be a
constant. It is an input energy source. It can be
a function of time, t and therefore varies with
t. Vo is a function of time, t. How do you obtain
Vo(t) ?
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Original DE
Rewriting
What does the DE tell you?
Tells you that the rate of change of the output
voltage is a function of the difference between
the input voltage Vi and output voltage, Vo.
If you solve the DE, you will be able to find the
output voltage, Vo as a function of time, t and
it will be written as V0(t)
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What then is the solution of a DE? How do you
solve for it?
A function y g(x) is a solution of a given DE
on some interval if y and its derivatives satisfy
the given DE.
A DE has many solutions.
Why is there a K?
Example
Differentiating the solution again
Since the solution admits any value of K, such
solutions containing arbitrary constants are
called general solutions of the DE. If the value
of K is fixed, then you get a particular solution.
Any solution obtained from the general solution
by giving specific values to the arbitrary
constants is called a particular solution of that
d.e., e.g. y sin x 1 is a particular solution
of y' cos x.
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Many solutions!
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General Solutions of First Order DE
Method 1 Separation of variables
A first order DE is separable if it can be
written in the form
Such a separable DE can be solved as follows
General Solution!
All the methods of integration that you have
learned before can be used to solve these
integrals.
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General solution involves K. K is then determined
using initial conditions.
Method 2 Use of Integrating Factors
For first order DE, the integrating factor method
is also quite effective and simple. An example
will be shown here.
Consider
Define a function
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Notice that R(t) is special because
Multiply both sides of the DE by R(t)
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Integrating both sides
Solution is a function of time, t
DE solved!
Function R(t) is called the integrating
factor! What does this integrating factor look
like for a DE with constant coefficient?
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Consider a first order DE from our RC circuit
example
Vi is the input voltage, C and R are constants,
and Vo is the output voltage.
Suppose Vi5 V, C1x10-6 F, R 1 MW, find the
output Vo. Assume that the capacitor is initially
uncharged.
This is a RC circuit driven by a constant DC
source of 5V. Vo is the output voltage across the
capacitor.
Above DE becomes
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compare
Solution
dummy variable, s
Solution
Integrating Factor
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What does this solution mean?
This is a solution that shows the variation of
the output voltage across the capacitor, Vo(t) as
time, t, changes.
All possible solutions as K is undetermined at
this stage. How can K be determined so that you
get a particular solution to the circuit problem?
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How is K determined? What must the solution
satisfy?
Solution must satisfy the physical condition at
the start of the experiment ie at time, t0
before the circuit was switched on!
Suppose we take two of the K values, K3 and K-5.
These solutions are
At time, t0, we have
Which is a more possible solution?
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This solution is not suitable because it gives,
Vo(0)8 V
This solution is more suitable because it gives,
Vo(0)0 V
What about time, t, goes to infinity?
Vo
5 V
Both solutions are reasonable as time goes to
infinity.
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We still have not found K, we have only guessed
the value of K. How do you find K?
Value of K should be determined by initial
conditions (IC).
Initial conditions Capacitor initially
uncharged and Vo(0)0 V.
General solution
From IC
Hence the particular solution is
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This is a typical charging profile of a capacitor
in a RC circuit. It takes about 5 sec to charge
up a capacitor to 5 V which is the input voltage.
Question If the R is larger, will it longer to
charge up the capacitor?
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How do you find the solutions of higher order DEs?
Consider
(1)
Time domain DE because in many engineering
problems, time domain is important. pi(t) are
continuous functions of t.
(2)
Consider
Let solution to (1) be yf(t) and solution to (2)
be yi(t).
Then y(t) yi(t) yf(t) is also a solution of
(1)!
Thus a general solution to (1) is of the form
very important in engineering!
y(t) yi(t) yf(t)
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In Engineering terms, r(t) is the external or
forcing input. Then yf(t) is the response to the
forcing input.
Equation (2) represents the behaviour when there
is no external forcing input. Hence yi(t) is the
natural response of the system.
Total behaviour, y(t), of the DE is determined by
initial conditions in the system as well as any
forcing signal, r(t), that is provided externally
in the system.
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Newtons Law of Cooling
T(t) will depend on the initial temperature of
the water. T0 is the ambient temperature and Klt0
because it is a cooling problem.
If there is external heat input
K1u(t) represents the rate of increase in the
temperature of the water in the cup.
Overall, T(t) depends on the initial temperature
of the water, the heat input, u(t) and the
ambient temperature, T0.
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Hence the solution to the DE which is due to the
initial conditions is given by
pi(t) real constants
Also known as the natural or zero response
Let this solution be given by yi(t).
The solution to the DE which is due to the
forcing signal is given by
Also known as the forced response
Let this solution be given by yf(t).
Then the total solution to the DE is given by
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Solution to the natural or zero response
This is the solution to the homogeneous equation
Consider the solution
Substitute this into the homogeneous DE.
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This equation is called the characteristic
equation.
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For 2nd order DE, there are 2 roots to the
characteristic equation. Depending on the roots,
the natural response is given by
Conjugate complex
For nth order DEs, there will be n roots. The
natural response will be combinations of the
solutions above. When there are repeated roots,
need to be careful.
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Example 1
Laws of Physics
Newtons second law of motion
Spring constant, k
y0
Mass, m
Hookes law for the spring
Displacement, y
Minus to show that it is an upward force ie
opposite to y
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Since the only force causing the motion is from
the spring force, we have
Homogeneous equation
Characteristic equation is
Purely imaginary roots!
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Conjugate complex
Solution to the spring problem
K1 and K2 are determined from initial conditions.
Assume that y(0)y0, y(0)0
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Assume y01, k1, m1.
What does this response mean?
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Questions to think about

• What is the effect of the initial velocity,
  y(0)?
• Is the response of y(t) realistic?
• If a constant force of 1N is now applied to the
  mass
• downwards, what do you expect to see?
• Formulate the problem?

What is the solution to a modified problem as
follows
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Solution to the forced response
This is the solution to the non-homogeneous
equation
The solution to this equation depends on the form
of r(t).
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For the same spring problem, suppose there is a
force acting on the mass which is given by
0.001t2 ie there is a force which is proportional
to t2. What does the y(t) look like?
Additional force term
Equation of motion
Let m1, k1
Homogeneous equation
Solution to HE obtained earlier
Non-homogeneous equation
Solve using table in previous slide.
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According to table, choose
Differentiating and substituting into the
non-homogeneous DE
Equating coefficients between the LHS and the RHS

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Total solution
Assume initial conditions y(0)1, y(0)0,
solve for K1, K2.
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Example RLC Circuit with a sinusoidal voltage
source
Inductor
Capacitor
Assume initial condition I(0)0, I(0)0
Power conservation
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Differentiating again
Dividing throughout by L
Natural response, Ii(t) from homogeneous DE
Forced response, If(t) from non-homogeneous DE
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Natural Response, Ii(t)
Assume a solution
Substitute into the homogeneous DE
If l1 and l2 are real and distinct
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If l1 l2 l ie repeated real
If l1 l jb, l2 l-jb ie complex conjugate
Forced Response, If(t)
Assume the general solution
Total solution
Try it and figure out if you can see what is
happening to current and voltage in the RLC
circuit! Have fun!