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

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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)

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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)
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)

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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 .
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