AMATH250 Lecture Notes  Partial Fraction Decomposition, Linear Equation, Initial Condition
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Published on 12 Apr 2013
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Algebraic equation:
Two numbers
make this an identity
t is independent variable

y is dependent variable

DE for unknown function
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 twoparameter family of solutions
State of motion in mechanics is given by and
Given initial conditions at :
and
what is
at ?
Introduction to Differential Equations (DE's)
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Example: Free Fall
is a given constant/parameter
General Solution
and
are arbitrary constants and they make this a general solution
It is a twoparameter family of solutions
Initial Conditions (IC's) at
Look at the general solution and find
and
in general solution to accommodate the IC's
Particular Solution
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
Drag
a)
Generalization of the model for motion in a field of gravity (2nd law of motion)
is the friction/drag coefficient always opposes motion
Changing Gravity
If is "large", force of gravity is given by
b)
This equation is called autonomous 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:
General Solution to DE
Contains arbitrary constants
Initial Conditions (IC)
What it sounds like: initial conditions that allow the general
solution to the DE to be fixed to a single result
Autonomous DE
The independent variable does not appear.
In this situation we can introduce a new dependent variable
for
, where
is the independent variable.
Classification of DE's
For some
(*)
is an order differential
equation. Generally nonlinear
Linear Differential Equation
Coefficients
are given functions of
as is the term
For
, this is a linear, nonhomogeneous, order
Differential Equation
If
then the DE is called homogeneous.
Modeling with DE's
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(*)
Explicit Solution
A function
that, when substituted for
in the DE (*), satisfies
this equation for all
is called an explicit solution.
Implicit Solution
A relation
, for some is said to be an
implicit solution to the differential equation
, for some on interval if it
defines one or more explicit solutions on
Solutions of order Differential Equations
(*)
for some
Subject to the initial condition
The Existence and Uniqueness Theorem
for the Initial Value Problem (IVP) (*)
If
and
are continuous functions on
that contains the point
then the IVP (*) has a unique solution
in some interval
Explicit Solution Example
Show that
is a general solution for
Implicit Solution Example
Show that relation for some positive constant is the implicit solution to
for
Use implicit differentiation assuming y is a function of x
defines a circle so there are two explicit solutions:
for
InitialValue problem for order DE
General solution
In general, there will be integration constants
Find to accommodate initial conditions:
Example
a)
Yes, for
b)
No unique solution. Both
are solutions
Does the DE
have a unique solution such that
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
Look for equilibrium solutions such that
. Solve
1)
For autonomous DE
Example
Solutions
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Document Summary
De for unknown function t is independent variable y is dependent variable where when substitute into de, we get identities for and. Example: newton"s second law is force, may depend on. For vertical motion of small object of mass , known. State of motion in mechanics is given by and. What it sounds like: initial conditions that allow the general solution to the de to be fixed to a single result. In this situation we can introduce a new dependent variable for. For some (*) is an order differential equation. Coefficients are given functions of as is the term. For , this is a linear, nonhomogeneous, order. General solution and are arbitrary constants and they make this a general solution. Look at the general solution and find and in general solution to accommodate the ic"s. Particular solution which satisfies both the original differential equation and the initial conditions. We solved an initial value problem (ivp) a de plus ic"s at.