Chapter day 4 Differential equations

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Chapter day 4Differential equations
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Recall from AP Calculus
The number of rabbits in a population increases
at a rate that is proportional to the number of
rabbits present (at least for awhile.)
So does any population of living creatures.
Other things that increase or decrease at a rate
proportional to the amount present include
radioactive material and money in an
interest-bearing account.
If the rate of change is proportional to the
amount present, the change can be modeled by
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Rate of change is proportional to the amount
present.
Divide both sides by y.
Integrate both sides.
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Integrate both sides.
Exponentiate both sides.
When multiplying like bases, add exponents. So
added exponents can be written as multiplication.
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Exponentiate both sides.
When multiplying like bases, add exponents. So
added exponents can be written as multiplication.
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So if we start with
We end with
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What if we have a series of differential
equations?

• dy1 ky1
• dt
• dy2 ky2
• dt
• dy3 ky3
• dt

We could solve of these individually y1
c1ekt y2 c2ekt y3 c3ekt Provided that we
have initial conditions for each of these to
solve for the constants
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If we define x

x1(t) x(t) x2(t)
xn (t)
This yields the equation x(t) Ax Which is easy
to solve in the case of a diagonal matrix.




x1 x2 x3
x1 x2 x3

• 0 0
• 0 -2 0
• 0 0 4

We can solve each of these as a separate
differential equation x1 3x1, x2
-2x2, x3 4x3 x1 (t) b1e3t, x2
(t) b2e-2t, x3 (t) b3e4t, This is the
general solution. We can solve for the constants
if given an initial condition.
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First order homogeneous linear system of
differential equations

• x1(t) a11x1 (t) a12 x2 (t) a1nxn (t)
• x2(t) a21x1 (t) a22 x2 (t) a2nxn (t)
• xn (t) an1x1 (t) an2 x2 (t) annxn (t)

We could write this in matrix form as
x1 (t) a11 a12 .
a12 x(t) x2 (t) A a21 a
22 a2n
xn (t)
an1 a n2 anm


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What if our system is not diagonal?
The system at the left can be written as du/dt
Au with a as

• du1 -u1 2 u2
• dt
• du2 u1 2u2
• dt

A -1 2 1 -2

How can we solve this system?
1 0
Initial condition u(0)
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• du/dt Au
• y eAt
• u(t) c1e? t x1 c2e ? t x2 cne? t xn
• Check that each piece solves the given system
• du/dt Au
• d (e? t x1) A e? t x1 ?e? t x1 A e?
  t x1
• dt ?x1
  Ax1

n
2
1
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Key Formulas

• Difference Equations
• Differential Equations

du/dt Au y eAt
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Solve the differential equations
The system at the left can be written as du/dt
Au with a as

• Start by computing the eigenvalues and
  eigenvectors

A -1 2 1 -2
What are the eigenvalues from inspection? Hint A
is singular The trace is -3
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Solve the differential equationsStep 1 find the
eigenvalues and eigenvectors
We can a solve via finding the determinant of A -
?I

• By inspection the matrix is singular therefore 0
  is an eigenvalue the trace is -3 therefore the
  other eigenvalue is -3

det -1-? 2 1 -2-?
Calculate the eigenvector associated with ?
0,-3


2 1
A -1 2 1 -2
For ? 0 find a basis for the kernel of A
For ? -3 find a basis for the kernel of A3I


1 -1
A 3I 2 2 1 1
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Solve the differential equations
The system at the left can be written as du/dt
Au with a as

• Note the solutions of the equations are going to
  be e raised to a power.

A -1 2 1 -2
The form that we are expecting for the answer is
y c1 e ? t x1 c2 e ? t x2
1
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The eigenvalues are already telling us about the
form of the solutions A negative eigenvalue will
mean that that portion goes to zero as x goes to
infinity. An eigenvalue of zero will mean that we
will have an e0 which will be a constant. We will
call this type of system a steady state.
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Solve the differential equations
Solve by plugging in eigenvalues into expected
equation and for ?1 and ?2. and the corresponding
eigenvectors in x1 and x2

A -1 2 1 -2

• y c1 e0t 2 c2 e -3t 1
• 1 -1

We find c1 and c2 by using the initial condition
Plugging in zero for t and the initial
conditions yields

Recall
1 0



Initial condition u(0)
1 c1 2 c2 1 0
1 -1
c1 1/3 c2 1/3
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Solve the differential equations

• The general solution is
• y 1/3 2 1/3 e -3t 1
• 1 -1

We are interested in hat happens as time goes to
infinity Recall our initial condition was 1
all of our quantity was in u1
0 Then as time
progressed there was flow from u1 to u2. As time
approaches infinity we end with the steady state
2/3
1/3


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• The solution to y ky is y y0ekt
• The solution to x Au
• is u c0eAt

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Applications of Linear Algebra

• Flow of water, electricity or money through a net
  work that continues over time.

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Applications of Differential Equations

• Differential Equations are the language in which
  the laws of nature are expressed. Understanding
  properties of solutions of differential equations
  is fundamental to much of contemporary science
  and engineering. Ordinary differential equations
  (ODE's) deal with functions of one variable,
  which can often be thought of as time
• - MIT

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http//en.wikipedia.org/wiki/Differential_equation

• Many fundamental laws of physics and chemistry
  can be formulated as differential equations. In
  biology and economics, differential equations are
  used to model the behavior of complex systems.
  The mathematical theory of differential equations
  first developed together with the sciences where
  the equations had originated and where the
  results found application. However, diverse
  problems, sometimes originating in quite distinct
  scientific fields, may give rise to identical
  differential equations. Whenever this happens,
  mathematical theory behind the equations can be
  viewed as a unifying principle behind diverse
  phenomena. As an example, consider propagation of
  light and sound in the atmosphere, and of waves
  on the surface of a pond. All of them may be
  described by the same second-order partial
  differential equation, the wave equation, which
  allows us to think of light and sound as forms of
  waves, much like familiar waves in the water.
  Conduction of heat, the theory of which was
  developed by Joseph Fourier, is governed by
  another second-order partial differential
  equation, the heat equation. It turned out that
  many diffusion processes, while seemingly
  different, are described by the same equation
  the BlackScholes equation in finance is, for
  instance, related to the heat equation.

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All of the following are stated in terms of
differential equations

• Newtons second law of dynamics
• Euler Lagrange theorem Classical mechanics
• Radioactive decay nuclear physics
• Newtons Law of cooling Thermodynamics
• Maxwells equation electro magnetism
• Einsteins field equation General relativity
• The Shroedinger equation Quantum mechanics
• Mathusian growth model Economics
• Verhulst equation biological population growth
• http//en.wikipedia.org/wiki/Differential_equation
  

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• http//ocw.mit.edu/courses/mathematics/18-03-diffe
  rential-equations-spring-2010/

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Homework wkst 8.4 1-9 odd, 2 and 8
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What if the matrix is not diagonal?

• White book p. 520 ex 3, 4, 5