M A : Ordinary Differential Equations

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1 M A : Ordinary Differential Equations Essential Class Notes & Graphics D 19 * Sections D07 D11 & D14 1

2 1. INTRODUCTION CLASS 1 ODE: Course s Overarching Functions An introduction to the subject (ODE) which is commonly considered the core of applied mathematics. A place to reinforce and extend your understanding of Calculus the next step after Calculus on the path that leads to real analysis.

3 Required Background 1. INTRODUCTION CLASS 1 The definition of the derivative, Derivative as slope, Conditions under which a derivative exists/does not exist, The rules of differentiation (including the Chain Rule), The derivative as a rate of change, The relationship between the 1 st derivative and the notions of increasing and decreasing functions Techniques of integration Differential Calculus (MA1021) Integral Calculus (MA1022)

4 1. INTRODUCTION CLASS 1 Techniques of Integrations! Keystone of the this discipline! Source:

5 1. INTRODUCTION CLASS 3 First-Order IVP Illustration by a Graph Solutions of the DE (x 0, y 0 ) y(x) I x

6 2. 1 st -ORDER DE CLASS 4 First-Order IVP Separation of Variables IVP:, Implicit solution: y(x) c = 4 x

7 2. 1 st -ORDER DE CLASS 5 First-Order Linear DE Integrating Factor IVP: C = General solution: y(x) x C = 0 C = C = -5

8 2. 1 st -ORDER DE CLASS 6 Models of Exponential Growth & Decay Example 1 Example 2 P A A 0 3P 0 A 0 /2 P 0 t ~ 24,180 t ~ 2.71 t (hours) t (years)

9 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 7 Second-Order IVP Illustration by a Graph Solutions of the DE y(x) (x 0, y 0 ) m = y 1 I x x 0

10 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 8 Homogeneous Eqs, Constant Coefficients, Case 1) IVP: Particular solution: Alternative ICs y (0) = 1, y (0) = 9 y(x) y (0) = 5, y (0) = 3 y (0) = 1, y (0) = 3 x

11 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 8 Homogeneous Eqs, Constant Coefficients, Case 2) IVP: Particular solution: y(x) y (0) = 3, y (0) = 7 y (0) = 1, y (0) = 7 Alternative ICs y (0) = 1, y (0) = 3 x

12 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 9 Homogeneous Eqs, Constant Coefficients, Case 3) IVP: Particular solution: y (0) = 3, y (0) = 3 y(x) Alternative ICs y (0) = 1, y (0) = 3 x y (0) = 1, y (0) = 2

13 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 12 Nonhomogeneous Eqs: Undetermined Coefficients Forms of a Particular Solution

14 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 13 Homogeneous Equations, Variation of Parameters IVP: Particular solution: Alternative ICs y (0) = 0, y (0) = 0 y (0) = 1, y (0) = 1 y(x) x

15 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 15 Flexible Spring Suspended from a Rigid Support Spring/Mass System (a) (b) (c) 1) Force due to gravity F g 2) Restoring force of the spring F s mechanical properties of the spring; resists stretching and trying to return the mass to its equilibrium 3) Damping force F d friction/resistant that the medium exerts on the mass x 4) External force F e possible additional force which may be applied trying to move the mass up/down

16 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 15 Periodic Functions of Harmonic Motion y 1 (t) = sin t & y 2 (t) = cos t y 2 (x) = cos t y(t) /4 /2 3 /4 5 /4 3 /2 7 /4 2 t y 1 (t) = sin t Period of motion: T = 2 / (Natural) frequency of motion: f = 1/T Circular frequency of motion: = k/m

17 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 16 Simple Harmonic Motion IVP: Equation of motion: x negative x positive x(t) (0, 2/3) Period /4 Amplitude A = 17/6 2/3 ~ /6 ~ t Frequency 4/

18 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 16 Simple Harmonic Motion Equation of Motion & Motion of the Mass POSITIONS: A B C etc. POSITIONS: A: The initial position of the mass below the equilibrium position B: The mass passing through the equilibrium C: The mass at its extreme displacement above the equilibrium etc.

19 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 16 Simple Harmonic Motion IVP: Equation of motion: x negative x positive x(t) (0, 1/4) Amplitude A = 17/16 Period 2 /16 Frequency 16/2 t

20 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 17 Unforced Damped Vibrations Overdamped Motion ( 2 2 > 0) IVP: Equation of motion: (0.157, 1.069) x(t) viscous medium t

21 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 18 Unforced Damped Vibrations Critically Damped Motion ( 2 2 = 0) IVP: Equation of motion: t x(t) viscous medium (1/4, 0.276)

22 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 18 Unforced Damped Vibrations Underdamped Motion ( 2 2 < 0) IVP: Equation of motion: k = 2, k = 10 viscous medium x(t) = 2, k = 17 = 2, k = 5 t

23 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 18 Forced Vibrations: Beats & Resonance Sinusoidal amplitude: Sinusoidal oscillation: Slow oscillations Rapid oscillations Beats Resonance The external force oscillates, but with t the amplitude of this oscillation grows infinitely large!

24 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 19 Forced Vibrations with Damping IVP: Equation of motion: transient solution steady-state solution equation of motion x(t) t steady-state solution transient solution

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