2021 Fall ME 6003 Applied Math ANNOTATED 092621

```Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
126
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
127
Higher Order DEs
Solve the nth degree polynomial:
If the roots are all real &amp; distinct,
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Lecture 01: Ordinary Differential Equations
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If the root m1 is of multiplicity k, the linear
independent solutions are:
And the general solution is a linear combination of the
above.
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Applied Mathematics
Lecture 01: Ordinary Differential Equations
129
Example 1.17
Solve:
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Lecture 01: Ordinary Differential Equations
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Undetermined Coefficients
To Solve:
We Must:
- Solve the Find the complimentary solution
-Find any particular solution of the nonhomogeneous equation
Assume:
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has the same form as
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Lecture 01: Ordinary Differential Equations
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Example 1.18
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Lecture 01: Ordinary Differential Equations
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Applied Mathematics
Lecture 01: Ordinary Differential Equations
133
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Applied Mathematics
Lecture 01: Ordinary Differential Equations
134
Example 1.19
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Lecture 01: Ordinary Differential Equations
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Lecture 01: Ordinary Differential Equations
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Applied Mathematics
Lecture 01: Ordinary Differential Equations
137
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Applied Mathematics
Lecture 01: Ordinary Differential Equations
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Variation of Parameters
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Lecture 01: Ordinary Differential Equations
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If y1 and y2 are linearly independent:
Variation of parameters:
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Lecture 01: Ordinary Differential Equations
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Lecture 01: Ordinary Differential Equations
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Lecture 01: Ordinary Differential Equations
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Lecture 01: Ordinary Differential Equations
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Applied Mathematics
Lecture 01: Ordinary Differential Equations
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Example 1.20
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Lecture 01: Ordinary Differential Equations
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Lecture 01: Ordinary Differential Equations
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Example 1.21
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Lecture 01: Ordinary Differential Equations
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Lecture 01: Ordinary Differential Equations
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Cauchy-Euler Equations
Trial Solution:
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Lecture 01: Ordinary Differential Equations
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Lecture 01: Ordinary Differential Equations
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Lecture 01: Ordinary Differential Equations
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Distinct, Real Roots:
Repeated Roots:
Complex Roots:
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Lecture 01: Ordinary Differential Equations
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Example 1.22
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Lecture 01: Ordinary Differential Equations
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Example 1.23
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Lecture 01: Ordinary Differential Equations
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Free Vibrations
Image: Kreyszig
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Lecture 01: Ordinary Differential Equations
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Hooke’s Law:
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Lecture 01: Ordinary Differential Equations
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Define:
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Lecture 01: Ordinary Differential Equations
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Lecture 01: Ordinary Differential Equations
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Period:
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Lecture 01: Ordinary Differential Equations
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Alternate Form
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Lecture 01: Ordinary Differential Equations
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Trig Identities
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Lecture 01: Ordinary Differential Equations
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Image: ZIll
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Lecture 01: Ordinary Differential Equations
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Image: Zill
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Lecture 01: Ordinary Differential Equations
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Example 1.24
An object weighing 2 lb stretches a spring 6 inches. At t =
0, the object is released from a point 8 inches below the
equilibrium position. Determine the equation of motion
for the object if its initial velocity is (a) 4/3 ft/s upward;
(b) 0; (c) 4/3 ft/s downward.
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
164
Fall 2021
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Lecture 01: Ordinary Differential Equations
165
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Lecture 01: Ordinary Differential Equations
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Lecture 01: Ordinary Differential Equations
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Lecture 01: Ordinary Differential Equations
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Damped Vibrations
Image: Zill
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Lecture 01: Ordinary Differential Equations
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Recall:
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Lecture 01: Ordinary Differential Equations
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Lecture 01: Ordinary Differential Equations
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Case I: Overdamping
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Lecture 01: Ordinary Differential Equations
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Case II: Critical Damping
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Lecture 01: Ordinary Differential Equations
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Example 1.25
An 8 lb weight stretches a spring 2 feet. The damping
coefficient is twice the instantaneous velocity. The weight
is released from its equilibrium position with an upward
velocity of 3 ft/s. Determine the equation of motion for
the object.
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Lecture 01: Ordinary Differential Equations
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Lecture 01: Ordinary Differential Equations
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Lecture 01: Ordinary Differential Equations
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Case III: Underdamping
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Example 1.26
An 16 lb weight is attached to a 5 foot spring. At
equilibrium, the spring measures 8.2 feet. If the weight is
pushed up and released from rest at a point 2 feet above
the equilibrium position, find the equation of motion for
the object ( c = 1lb-s/ft).
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Applied Mathematics
Lecture 01: Ordinary Differential Equations
179
Fall 2021
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Lecture 01: Ordinary Differential Equations
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Lecture 01: Ordinary Differential Equations
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Lecture 01: Ordinary Differential Equations
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Beam Deflection
Image: Zill
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Lecture 01: Ordinary Differential Equations
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Image: Zill
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Lecture 01: Ordinary Differential Equations
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Lecture 01: Ordinary Differential Equations
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Boundary Conditions
Image: Zill
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Lecture 01: Ordinary Differential Equations
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Example 1.27
A beam of length 1 m is fixed at both ends. Find the
deflection of the beam if a constant load is uniformly
distributed along its’ length.
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
187
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
188
Fall 2021
Applied Mathematics
Lecture 01: Ordinary Differential Equations
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Lecture 01: Ordinary Differential Equations
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Lecture 01: Ordinary Differential Equations
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