M A : Ordinary Differential Equations

Transcription

1 M A : Ordinary Differential Equations Essential Class Notes & Graphics C 17 * Sections C11-C18, C

2 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)

3 1. INTRODUCTION CLASS 5 First-Order Linear DE Integrating Factor C = General solution: y(x) x C = 0 C = C = -5

4 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 8 Homogeneous Equations, Constant Coeff., Case 1) 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

5 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 8 Homogeneous Equations, Constant Coeff., Case 2) 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

6 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 9 Homogeneous Equations, Constant Coeff., Case 3) 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

7 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 12 Nonhomogeneous Equations: Undetermined Coeffs Forms of a Particular Solution

8 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 13 Homogeneous Equations, Variation of Parameters Particular solution: Alternative ICs y (0) = 0, y (0) = 0 y (0) = 1, y (0) = 1 y(x) x

9 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

10 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 16 Simple Harmonic Motion x negative x positive Equation of motion: (0, 2/3) Amplitude A = 17/6 2/3 ~ /6 ~ x(t) t Period p/4 Frequency 4/p

11 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.

12 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 16 Simple Harmonic Motion x negative x positive Equation of motion: x(t) (0, 1/4) Amplitude A = 17/16 t Period 2p/16 Frequency 16/2p

13 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 17 Unforced Damped Vibrations Overdamped Motion (l 2 w 2 > 0) Equation of motion: (0.157, 1.069) x(t) viscous medium t

14 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 18 Unforced Damped Vibrations Critically Damped Motion (l 2 w 2 = 0) Equation of motion: t x(t) viscous medium (1/4, 0.276)

15 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 18 Unforced Damped Vibrations Underdamped Motion (l 2 w 2 < 0) b Equation of motion: k b = 2, k = 10 viscous medium x(t) b = 2, k = 17 b = 2, k = 5 t

16 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!

17 3. 2 nd -ORDER LINEAR EQUATIONS CLASS 19 Forced Vibrations with Damping Equation of motion: transient solution steady-state solution equation of motion x(t) t transient solution steady-state solution

18 3. THE LAPLACE TRANSFORM CLASS 23 IVP by the Laplace Transform Solution: Solution of the IVP y = 8e 3t y(t) t IVP s r.h.s.: y = 13sin2t

19 3. THE LAPLACE TRANSFORM CLASS 23 IVP by the Laplace Transform Solution: y(t) Solution of the IVP y = e 4t t IVP s r.h.s.: y = e 4t

20 3. THE LAPLACE TRANSFORM CLASS 24 IVP by the Laplace Transform Solution: Solution of the IVP y(t) y = 2e 3t t IVP s r.h.s.: y = t 2 e 3t

21 3. THE LAPLACE TRANSFORM CLASS 26 IVP Involving Piecewise-Defined Functions 1, 0 < t < 1 y + y = f(t), y(0) = 0, y (0) = 0, where f(t) = 0, t > 1 1 cost, 0 < t < 1 Solution: y(t) = 1 cost U(t 1)[1 cos(t 1 )] = 0.46cost sint, t > 1 f(t) y(t) t y(t) = 1 cost y(t) = 0.46cost sint

22 3. THE LAPLACE TRANSFORM CLASS 26 IVP Involving Piecewise-Defined Functions y + y = f(t), y(0) = 5, where f(t) = 0, 0 < t < p 3cost, t > p Solution: y(t) = 5e t + (3/2) e (t p) U(t p) (3/2)sin(t p)u(t p) (3/2)cos(t p)u(t p) = 5e t, 0 < t < p = 5e t (3/2)e (t p) + (3/2)sint + (3/2)cost, t > p y(t) = 5e -t y(t) = 5e -t + (3/2)e -(t-p) + (3/2)sint + (3/2)cost y(t) f(t) p t

23 Conclusion Considered material: General intro First-order equations Separation of variables Integrating factor Second-order linear equations Homogeneous: method of characteristic equation Nonhomogeneous: undetermined coefficients and variation of parameters The Laplace Transform This course is a brief introduction to the subject matter. If you need to do some work in mathematical modeling, MA2051 may be a good starting point for your deeper work in this realm.