Chapter 1: Introduction to Differential Equations

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Chapter 1 Introduction to Differential Equations
1.1 1.2 1.3 Definitions and Terminology Initial Value Problem DE as Mathematical Models
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Sec 1.1 Definitions and Terminology
Definition 1.1 Differential Equation
An equation containing the derivatives of one or
more dependent variables with respect to one or
more independent variables, is said to be a
differential equation (DE)
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Sec 1.1 Definitions and Terminology
Classification
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Classification By Type (ODE,PDE)
Classification By Type
If an equation containing only ordinary
derivatives ? it is said to be Ordinary
Differential Equation (ODE)
An equation involving partial derivatives ? it
is said to be Partial Differential Equation (PDE)
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Classification By Order
The order of a differential equation (ODE or PDE)
is the order of the highest derivative highest
derivative in the equation.
Classification By Order
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Classification By Order
Write the DE
where
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Classification By Linearity
Classification By Linearity
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Classification By Linearity
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Classification
ODE or PDE
highest derivative
Linear in y,y,y,
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System of DE
System of two Ordinary Differential Equations 2ed
order, linear, ODE
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Navier-Stokes Equations

• ODE or PDE
• order??
• linear or nonlinear

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Sec 1.1 Definitions and Terminology
HW
HW
HW
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Sec 1.1 Definitions and Terminology
Definition 1.2 Solution of an ODE
Any function defined on an interval I and
possessing at least n derivatives that are
continuous on I, which when substituted into an
nth-order ODE reduces the equation to an
identity, is said to be a solution of the
equation on the interval.
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Sec 1.1 Definitions and Terminology
Definition 1.2 Solution of an ODE
Any function defined on an interval I and
possessing at least n derivatives that are
continuous on I, which when substituted into an
nth-order ODE reduces the equation to an
identity, is said to be a solution of the
equation on the interval.
Interval of existence interval of validity
domain of the solution
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Sec 1.1 Definitions and Terminology
Families of Solutions
Particular Solution A solution of DE that is
free of arbitrary parameters
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Sec 1.1 Definitions and Terminology
Not a member of the family no value for c
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Office Hours
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Sec 1.1 Definitions and Terminology
Implicit Solution
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Sec 1.1 Homework Problems
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Sec 1.1 Homework Problems