1 y and x 2 y Obtain dynamical system x 1 x 2 x 2 f t x 1 x 2 The state

1 y and x 2 y obtain dynamical system x 1 x 2 x 2 f t

This preview shows page 6 - 10 out of 32 pages.

1 = y and x 2 = y 0 Obtain dynamical system ˙ x 1 = x 2 ˙ x 2 = f ( t, x 1 , x 2 ) The state variables are y and y 0 , which have solutions producing trajectories or orbits in the phase plane For movement of a particle, one can think of the DE governing the dynamics produces by Newton’s Law of motion and the phase plane orbits show the position and velocity of the particle Joseph M. Mahaffy, h [email protected] i Lecture Notes – Second Order Linear Equations — (6/32)
Image of page 6

Subscribe to view the full document.

Introduction Theory for 2 nd Order DEs Linear Constant Coefficient DEs Second Order Differential Equation Dynamical system formulation Classic Examples Classic Examples Spring Problem with mass m position y ( t ), k spring constant, γ viscous damping, and external force F ( t ) Unforced, undamped oscillator, my 00 + ky = 0 Unforced, damped oscillator, my 00 + γy 0 + ky = 0 Forced, undamped oscillator, my 00 + ky = F ( t ) Forced, undamped oscillator, my 00 + γy 0 + ky = F ( t ) Pendulum Problem - mass m , drag c , length L , γ = c mL , ω 2 = g L , angle θ ( t ) Nonlinear, θ 00 + γθ 0 + ω 2 sin( θ ) = 0 Linearized, θ 00 + γθ 0 + ω 2 θ = 0 RLC Circuit Let R be the resistance (ohms), C be capacitance (farads), L be inductance (henries), e ( t ) be impressed voltage Kirchhoff’s Law for q ( t ), charge on the capacitor Lq 00 + Rq 0 + q C = e ( t ) , Joseph M. Mahaffy, h [email protected] i Lecture Notes – Second Order Linear Equations — (7/32)
Image of page 7
Introduction Theory for 2 nd Order DEs Linear Constant Coefficient DEs Existence and Uniqueness Linear Operators and Superposition Wronskian and Fundamental Set of Solutions Existence and Uniqueness Theorem (Existence and Uniqueness) Let p ( t ) , q ( t ) , and g ( t ) be continuous on an open interval I , let t 0 I , and let y 0 and y 1 be given numbers. Then there exists a unique solution y = φ ( t ) of the 2 nd order differential equation: y 00 + p ( t ) y 0 + q ( t ) y = g ( t ) , that satisfies the initial conditions y ( t 0 ) = y 0 and y 0 ( t 0 ) = y 1 . This unique solution exists throughout the interval I . Joseph M. Mahaffy, h [email protected] i Lecture Notes – Second Order Linear Equations — (8/32)
Image of page 8

Subscribe to view the full document.

Introduction Theory for 2 nd Order DEs Linear Constant Coefficient DEs Existence and Uniqueness Linear Operators and Superposition Wronskian and Fundamental Set of Solutions Linear Operator Theorem (Linear Differential Operator) Let L satisfy L [ y ] = y 00 + py 0 + qy , where p and q are continuous functions on an interval I . If y 1 and y 2 are twice continuously differentiable functions on I and c 1 and c 2 are constants, then L [ c 1 y 1 + c 2 y 2 ] = c 1 L [ y 1 ] + c 2 L [ y 2 ] . Proof uses linearity of differentiation. Theorem (Principle of Superposition) Let L [ y ] = y 00 + py 0 + qy , where p and q are continuous functions on an interval I . If y 1 and y 2 are two solutions of L [ y ] = 0 ( homogeneous equation ), then the linear combination y = c 1 y 1 + c 2 y 2 is also a solution for any constants c 1 and c 2 .
Image of page 9
Image of page 10
  • Fall '08
  • staff

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask You can ask ( soon) You can ask (will expire )
Answers in as fast as 15 minutes