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University of Waterloo
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Applied Mathematics
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AMATH 250
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Julia Roberts
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Lecture

School

University of Waterloo
Department

Applied Mathematics

Course Code

AMATH 250

Professor

Julia Roberts

Description

Introduction to Differential Equations (DE's)
September-10-12
2:01 PM
Algebraic equation:
Two numbers make this an identity
DE for unknown function
- t is independent variable
- y is dependent variable
where and
when substitute into DE, we get identities for
Example: Newton's Second Law
is force, may depend on
For vertical motion of small object of mass , known
Free fall:
Solve the DE
Solve by using antidifferentiation
This is a two-parameter family of solutions
State of motion in mechanics is given by and
Given initialconditions at : and what is at ?
AMATH 251 Page 1 Modeling with DE's
September-12-12 Example: Free Fall
1:39 PM
is a given constant/parameter
General Solution to DE
Contains arbitrary constants General Solution
Initial Conditions (IC)
and are arbitrary constants and they make this a general solution
What it sounds like: initial conditions that allow the general It is a two-parameter family of solutions
solution to the DE to be fixed to a single result
Autonomous DE Initial Conditions (IC's) at
The independent variable does not appear.
In this situation we can introduce a new dependent variable
Look at the general solution and find and in general solution to accommodate the IC's
for , where is the independent variable.
Classification of DE's
For some
(*) is an order differential Particular Solution
equation. Generally non-linear
which satisfies both the original differential equation and the initial conditions.
As such, it predicts at any and at
We solved an Initial Value Problem (IVP) — a DE plus IC's
Linear Differential Equation
Drag
Coefficients are given functions of as is the term Generalization of the model for motion in a field of gravity (2nd law of motion)
For , this is anear, non-homogeneous, order a)
Differential Equation is the friction/drag coefficient —always opposes motion
If then the DE is calledhomogeneous. Changing Gravity
If is "large", force of gravity is given by
b)
This equation is calledautonomous since the independent variable does not appear, only its
derivatives.
For a)
Introduce new dependent variable and write
So we now have a system of two first order DE's
For b)
Rocket burning fuel
Rocket burning fuel, expelled at at velocity relative to rocket
Mass of rocket:
AMATH 251 Page 2 Solutions Explicit Solution Example
September-14-12
1:34 PM
Show that
(*)
is a general solution for
Explicit Solution
A function that, when substituted for in the DE (*), satisfies
this equation for all is called anexplicit solution.
Implicit Solution Example
Implicit Solution Show that relation for some positive constant is the implicit solution to
A relation , for some is said to be an
implicit solution to the differential equation for
, for some on interval if it Use implicit differentiation assumingy is a function of x
defines one or more explicit solutions on
Solutions of order Differential Equations
(*) for some defines a circle so there are two explicit solutions:
Subject to the initial condition for
The Existence and Uniqueness Theorem Initial-Value problem for order DE
for the Initial Value Problem (IVP) (*)
If and are continuous functions on General solution
that contains the point In general, there will be integration constants
then the IVP (*) has a unique solution in some interval
Find to accommodate initial conditions:
Example
Does the DE have a unique solution such that
a)
Yes, for
b)
No unique solution. Both are solutions
General solution
Solution Methods
Direction Fields for 1st order DE's
, we can generate a picture of the family of solution curves that correspond
to general solutions.
Look for isoclines with constant slope
Example
Rays exiting from the circle are isoclines.
Autonomous Equations
For autonomous DE
1) Look for equilibriumsolutions such that . Solve
Example
AMATH 251 Page 3 Example
Euler's Method
September-17-12 Euler's Method for
1:34 PM
Euler's Exact
st 0 1 4 4
Numerical Approximation for 1 -order DE's
Euler's Method 1 1.1 4.200 4.213
2 1.2 4.425 4.452
3 1.3 4.678 4.720
Find approximationfor exact solution 4 1.4 4.959 5.018
5 1.5 5.271 5.318
Use partitionof t-axis
Example
Linear approximation:
Evaluate using Euler's method
Exact:
Approximation:
1 1 2
2 0.5 2.250
4 0.25 2.441
8 0.125 2.566
16 0.0625 2.638 Dimensions of PhysicalQuantities Free Fall
September-17-12
2:02 PM
In mechanics, all physical quantities have dimensions of form
← M ss
← L
← Time
Consistency Requirements
1) Dimensional Homogeneity of equation:
Work
May only add, subtract or equate quantities that have the same
dimension.
2) Quantities havingdifferent dimensions may only be combined by
multiplicationof division
Notation for quantity has dimension
Free fall with Drag
Dimension of drag coefficient
Dimensionless time Using dimensionless time:
notice
Define dimensionless time: Solving First Order DEs
September-19-12 Separable DE
1:25 PM Let
st
Types of 1 Order DE's
That can be solved analytically L
Solution is given by
1) Separable Equation In general this is implicit
2) Linear Equation Example
3) Exact Equation
Example
Recall example of a body experiencing gravity and drag force
Define dimensionless time
Define terminal velocity
so that
Define
Stable equilibrium solution where
Population Growth
y = number of species at time with
Simple equation of growth with rate
K is the carrying capacity of the system
This is an autonomous equation, there is no in
Equilibrium solution:
Stable equilibriums have unstable have
Solve by separable variables
Partial fraction decomposition
AMATH 251 Page 6 Partial fraction decomposition
s
where
Initial Condition
Second Order Des with missing independent variable
s
Let and use chain rule.
Chain rule:
In mechanics, separable DE
AMATH 251 Page 7 More Solving In mechanics, Newton in 1-D
September-21-12 , missing independent variable t
1:37 PM
Existence and Uniqueness
y "independent"
Theorem
s
and
If and are continuous in
interval I containing the initial point
then there exists a unique solution to
IVP for all
Conservation of energy
IC at
Solve for
Sub into 1
Solve as separable DE
Escape velocity
Minimum speed to send rocket to space
escape velocity:
s
Cases of DE's that may be converted to separable DE's
Homogeneous
1) Example
Use substitution
Assignment Q1
2)
Define
Linear Equations
given function of t
IC
Solve by Method of Integrating Factors
(see problem 39 in Sec.1.2, page 26 for Method of Variation of Parameters)
Bernoulli's equation Bernoulli's equation
Define
Assignment Q2 Exact Solution Proof
Exact Differential Equations
September-24-12 If is a solution to (*) then upon substitution into , use Chain rule
1:28 PM
Exact Equation
(*)
This differential equation is called an exact equation
if there exists a that is such that
Proof of Test of Exactness
Its general solution is given by Show f exact, then there exist F, and
Differentiate
Test of Exactness
Let be
continuous functions on a simply-connected domain
in
Since and are continuous
Then is an exact equation
if and only if in Other way
Assume and
Integrate with respect to , keep fixed.
Define such that
Use this to determine
This must be independent of , derivative with respect to must vanish.
Example
Solve the differentiation equation
Test for exactness
Integrate
Putting together
Initial condition: gives
AMATH 251 Page 10 Second Order Differential Equations Modelling with 2nd -order DE's with constant coefficients
September-24-12 Mechanical sprint -mass oscillator
2:10 PM
Standard Form of Linear
(Non-HomogeneousDE)
Giventhethreefunctions of
Standard Form of Linear With Constant Coefficients external force
are constant restoring force of spring
drag force
Origin is the equilibrium position (unstretched spring)
s s
Displacementofthe mass fromequilibriumpositionfollows Newton
s
Linearization of Pendulum
Linear Displacement
Forcesactingtangentially
s
s
B s s s
Forsmallangulardisplacements:
s
Dropall powersbut theleadingone
s
Initial Conditions
Electrical Oscillator: Series RLC Circuit
AMATH 251 Page 11 s s
State of the RLC circuit is defined by
s
s
s
Kirchhoff's Voltage Law
Initial Conditions
Equivalent DE for Current
Differentiate
Initial Conditions
To get , set in
AMATH 251 Page 12 How to represent every (general) solution to 2 ndorder homogeneous linear DE
Theory of order linear DE's
September-26-12 A) Need and linearly independent
2:08 PM
Forget solutions to (*) which are identically zero on (trivial solutions)
Note
If is a complex-valued solution of DE (*) with real-valued coefficients,
Initial Conditions so are its real , and imaginary parts.
Existence and Uniqueness Theorem Proof of Theorem (Solution to IVP)
If and are continuous functions on some interval ,
which contains , then there exists a unique solution to the
differential equation for all ; which satisfies the initial
conditions.
Linear Operator
Define linear operator
← s
← H s Proof of Theorem
1) Assume
Theorem
If and are sufficiently differentiable on the interval and 2) . Assume that for some
and are any constants then
Corollary (Superposition Principle) s
If and are any solutions of the homogeneous DE
then the linear combination is also a solution to that
same DE. Proof of Abel's Theorem
Linear Independence For
Two functions and are said to be linearly independent on
interval iff neither is a constant multiple of the other.
If and are solutions to then
Fundamental Set (Basis)
If and are solutions of the homogeneous DE (*) that are linearly is given by
independent on then they are said to form a fundamental set, or
basis, of solutions.
Proof
Theorem: Representation of General Solution to (*)
Representation of general solutions to homogeneous, linear, second-
order differential equation.
If and are linearly independent solutions of linear
on , then every solution of that equation is give by
where and are arbitrary constants that may be determined from
the
Comment
are arbitrary.
Wronskian
The Wronskian of two functions and is defined by
Abel's Theorem for Wronskian
For any two solutions of the DE (*), and
then
Corollary
is either never zero or always zero on
so the value of depends on point
Theorem
If and are any two solutions of on , then they are
linearly dependent on iff their Wronskian is identically zero on .
AMATH 251 Page 13 Example
Solving order DEs with constant coefficients Solve the IVP for
October-01-12
1:32 PM
Characteristic equation
General solution:
Fundamental Solutions
Find the fundamental solutions to
s s
and
Assume solution in form with being a parameter
and the solution is
Solve the characteristic equation (auxiliary equation):
Example
These two roots give us
These are linearly independent iff
1) Distinct real roots, s s
2) Distinct complex roots
s s
s s s s
s s s
3) Equal roots,
We only have
Use reduction of order, assuming
s s
Alternate method
How can we write this as
Assume and let s
General solution:
s s s s s
Expand with Maclaurin series
s s s
and depend on so that
s
Example
s
Conclusion
If repeats the just multiply by a single factor of
Non-Homogeneous Linear Equation
Let be general solution of associated linear
equation: e.g.
and let be any particular solution of the non-homogeneous equation
Then the general solution of the equation is
AMATH 251 Page 14 Superposition Principle for Non-Homogenous Linear DEs
October-03-12
1:26 PM
Example
General Solution to Linear Second Order Non-Homogeneous DEs Find to
Let are undetermined coefficients
general solution of associated homogeneous equation
s Since and are linearly independent functions
Then the general solution of is
Comment
If and and solve and Example
then solves
Method of Undetermined Coefficients
DE with constant coefficient
s
If is
• polynomial in
• exponential Example
• s or s
• or product of the above
Use Method of Undetermined Coefficients
1) , assume What went wrong?
2) assume Recall: linearly independent solutions of the associated homogeneous DE
3) s or s assume were
For , assume
Exception
If reproduces any of the functions in the basis of solutions to
the homogeneous DE, then just multiply your assumption for by a
single factor of .
Method of Variation of Parameters (or Constants)
for
Example
If we have a fundamental set of solutions Solve IVP for
Recall
To find particular solution
Assume
Need to determine functions
Assume
Sub into initial DE
1)
2) Example
s Recall
Example
s
s s
s s
Quiescent Initial Conditions s s
Compare solutions to the IVPs
1)
s s s
2)
s s
Difference is solutions of the IVP
AMATH 251 Page 15 2)
s s
Difference is solutions of the IVP
With 0 for initial conditions, called Quiescent initial conditions.
Example of Method of Variation of Parameters
Find for s
From
s s
s s s s
s s
s
Solution to the IVP with Quiescent initial conditions.
s
Integrate formulas for such that s
s s
s
s
s s s
Example - Multiplication by t for constant coefficients
Find for
Green's Propagator
Recall
Use method of variation of parameters
serves in some sense as an inverse of
but is linearly dependent on so
AMATH 251 Page 16 OscillatorDE and Resonance
October-10-12
1:31 PM
Recall: mechanical and electrical oscillators
• Mass on spring, possiblydamped
• RLC circuit
Both systems are describedby
where or
= natural frequency
= damping parameter. Normally
Assume harmonicforcing
s
F = amplitude,
= frequency
Free Oscillations
First consider free oscillationswith
The associatedhomogeneousDE
Aside: we could use dimensionlesstime
Determinantcases:
1) : overdamped motion of oscillator
2) : critically damped oscillator
3) : underdamped oscillator
s s
s
and are determined by initial conditions:
Forced Oscillations
s
General solution:
s s s s
Undeterminedamplitude, phase shift
Steady-StateResponse
Since persists, is called steady-stateresponse.
Transient Response
Since , it is called the transientresponseof the system.
The initialconditionsare "forgotten".
Solving
To determine and assume complex-valued solution
s
AMATH 251 Page 17 s
s
Find particular solution and take
Assume where
s
Resonance Amplitude
s s
s
s
Solution to Forced Oscillations
s
Remark: Notation in textbook
Analyze this as function of "reduced" frequency
Resonance
For , there is a local maximum in the plot of .
This is the resonantfrequency.
If
Zero Damping
s
s s
Persistsfor all t
s s
s
Recall
s Resonance Phase
General solution
s
What happens as s
"Borrow" part from and consider quiescentsystem
s
s s
s
s s
Beats
consistsof a large amplitude and large period sine wave filled with a s
small amplitude wave. These are known as beats
s
s
This is a linearly-increasingamplitude sine wave.
AMATH 251 Page 18 s
This is a linearly-increasingamplitude sine wave.
AMATH 251 Page 19 Systems of First Order DEs
October-15-12 Homogeneous Undamped Oscillator
1:55 PM Solve IVM for homogeneousequation
Phase Portrait Characteristic equation
The phase portrait of the solution is the description of a
s s
solution to the DE as a circle on theand plane.
s s
s s
s s
The vector defines the state of the system at time
s s
s s
Describe the state for the system by a curve in the plane.
Rewrite as a system of equations for state variables
Define the vector
so the system becomes
This looks like a separable equation
Matrix exponential:
s s
s s
s
s
where s
This is a circle on the plane. Phase portrait
Recall
Multiply by and obtain
This equation represents conservation of energy. Laplace Transform ConsidertheoscillatorDE
periodicforcing
October-17-12
1:38 PM Example:
Integral Transform
An IntegralTransformis a linear operatorthat maps
functions tofunctions , definedby
is called the Kernel
Laplace Transform
Laplace transform of , defined on is the
function defined by
How do we solve this?
1) Undetermined coefficients
forsolutionsoverintervalswhen is continuous
Note 2) The Green'sfunction
In thiscourse, is realbut in general it is possiblethat
What about higher-order linear DE's with constant coefficients ?
Notation These arise with coupled oscillators.
The domain of definition of is the set of alvalues for 3) Can use undeterminedcoefficients
which the integral exists (converges).
Note:since involvesan improperintegral
This has complex roo

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