Impulse forcing

Impulse force is an external force acting on a harmonic oscillator which is very large at one moment and is zero any other time.

 very large, t = a; g(t) = 0, t 6= a.

δ-function (Delta function)  ∞, t = a; Z ∞ δa(t) = , and δa(t)dt = 1. 0, t 6= a. −∞

−as Formulas: L[δa(t)] = e , L[δ0(t)] = 1.

L[δa(t)] = sL[ua(t)]: Delta function is the derivative of Heaviside function.

00 0 0 Example 11: y + 2y + 5y = δ3(t), y(0) = 1, y (0) = 1. 00 0 Example 12: y + 4y = δ4(t), y(0) = 0, y (0) = 0. Qualitative behavior: Before the impulse, the equation is “normal” at the impulse, the derivative of the solution has a sudden change due to the impulse force, but the solution is still continuous.

Similarity: 0 0 y = ua(t) (continuous not differentiable), y = δa(t) (not continuous at t = a) 00 00 y = ua(t) (continuous and differentiable), y = δa(t) (continuous not differentiable) Qualitative behavior

00 0 Example 13: y + 4y = u5(t) sin(3(x − 5)), y(0) = 2, y (0) = 0. Solution: y(t) = 2 cos(2t) + 0.3u5(t) sin(2(t − 5)) − 0.2u5(t) sin(3(t − 5)). Qualitative: the external force is continuous, so y, y 0 and y 00 are all continuous, and y 000 is discontinuous at t = 5.

00 0 0 Example 14 y + 2y + 5y = δ3(t), y(0) = 1, y (0) = 1. Solution: −t −t −(t−3) y(t) = e cos(2t) + e sin(2t) + 0.5u3(t)e sin(2(t − 3)) Qualitative: the external force is impulsive, so y 00 is impulsive, y 0 is discontinuous, and y is continuous Final Exam

Final Exam: Dec. 13 (Friday), 9-12am, this classroom (Jones 306) any change must be approved by Dean of Students 3 sample exams are at webpage and solution keys are at Blackboard website

Final Exam Office Hours: Dec. 11 (Wednesday) 4-6pm, Jones Hall 100B Dec. 12 (Thursday) 10-12am, 3-6pm, Jones Hall 100B

Composition of the exam:

Chapter 1 ≈ 25%, Chapter 2-3 ≈ 25% Chapter 4-5 ≈ 25%, Chapter 6 ≈ 25%

About 12 − 15 short problems, 4 − 5 longer problems. similar to Fall 2003, Fall 2004, Fall 2016 final exams (see Webpage/Blackboard)

A table of Laplace transform will be given to you in exam (page 626 of textbook) But not other formula sheets (trigonometric formulas, integrals, derivatives) Chapter 1

1 Analytic methods: separation of variables, linear equation (integrating factor, homogeneous/nonhomogeneous) 2 Numerical method: Euler’s method for scalar equations 3 Qualitative methods: slope field, graph of solutions, phase line, equilibrium points and their stability (sink, source, node), asymptotic behavior of the solutions, bifurcation, linearization; 4 Modeling: population models (Malthus, logistic, logistic with harvesting), mixing problems (constant volume, varying volume), banking problems (saving, withdrawing, mortgage); 5 Existence, uniqueness and the defining domain of the solution. Chapter 2-3

1 Analytic methods: vector form of system, converting 2nd order equation to 1st order system, equilibrium solutions for systems, decoupled systems, eigenvalues and eigenvectors for 2 × 2 matrix, linear principle, solve linear systems (first order, two-variables) (eigenvalues, eigenvectors) det(A − λI ) = 0, (A − λ1)V1 = 0, (A − λ2)V2 = 0 λ t λ t Real eigenvalues: Y(t) = c1e 1 V1 + c2e 2 V2 λ t Complex eigenvalues: Y(t) = e 1 V1 = Yr + iYi , Y (t) = c1Yr + c2Yi λ t λ t Repeated eigenvalues: Y(t) = c1e 1 V1 + c2e 2 (tV1 + V2) (A − λI )V2 = V1. 2 Qualitative methods: phase plane, phase portrait, solution curve, phase portrait of linear systems, straight line solutions, types of linear systems, asymptotic behavior of solutions of linear systems, clockwise or counterclockwise Type of linear systems: sink, source, saddle, spiral sink, spiral source, degenerate sink(source), trying to spiral sink(source), center, star sink(source), parallel lines 3 Modeling: predator-prey system, mass-spring system, linear systems. Chapter 4-5 and section 3.6

Summary of Chapter 4: y 00 + py 0 + qy = f (t)

1 Analytic methods: solution of homogeneous and inhomogeneous equations, undetermined coefficient method, frequency of beats 00 0 2 λ t λ t y + py + qy = 0: λ + pλ + q = 0, y = c1e 1 + c2e 2 00 0 λ t λ t y + py + qy = f (t): y = c1e 1 + c2e 2 + yp(t) at at at particular solution yp: e 7→ ke or kte , sin(at) (or cos(at)) 2 2 7→ k1 sin(at) + k2 cos(at), at + bt + c 7→ At + Bt + C. 2 Qualitative methods: underdamped, overdamped and critically damped oscillations, asymptotic behavior of solutions (harmonic oscillators), resonance, beats. 3 Models: Forced harmonic oscillators Summary of Chapter 5: x0 = f (x, y), y 0 = g(x, y)

1 Analytic methods: find nullclines, equilibrium points, Hamiltonian systems and dissipative systems 2 Qualitative methods: nullclines, equilibrium points, direction of vector field, Jacobian, linearization, bifurcation. 3 Models: competition model, cooperative model, pendulum Chapter 6

y 0 + qy = f (t), y(0) = a, y 00 + py 0 + qy = f (t), y(0) = a, y 0(0) = b Z ∞ 1 Definition of Laplace transform L[y(t)] = Y (s) = y(t)e−st dt, Heaviside 0 function and piecewise defined functions, Dirac-delta function 2 Methods to solve inverse transform: partial fractions, completing the square −as 3 Shifting formulas: if L[f (t)] = F (s), then L[ua(t)f (t − a)] = e F (s) and L[eat f (t)] = F (s − a). 4 A table of Laplace transform will be given to you in exam (page 626 of textbook)