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Unformatted text preview: 3.1/3.4/3.5: HOMOGENEOUS EQUATIONS WITH CONSTANT COEFFICIENTS KIAM HEONG KWA A general second-order differential equation has the form (1) d 2 y dt 2 = f t,y, dy dt , where f is a given function of three variables. Second-order equations arise very often in applications. The most well-known second-order equation is Newtons second law of motion m d 2 y dt 2 = F t,y, dy dt that governs the trajectory of a particle of mass m moving in response to a force F . In the last equation, y and dy dt denote the position and the velocity of the particle at time t respectively. A second-order initial- value problem (IVP) consists of a differential equation such as (1) and a pair of initial conditions (ICs) (2) y ( t ) = y , y ( t ) = y o at a given point t = t . Here y and y are some prescribed values of the unknown function y and its derivative. A second-order equation such as (1) is said to be linear provided the function f has the form f t,y, dy dt =- p ( t ) dy dt- q ( t ) y + g ( t ) for some functions p ( t ), q ( t ), and g ( t ). It is generally written in the standard form (3) y 00 + p ( t ) y + q ( t ) y = g ( t ) . It is easy to verify the principle of superposition which says that if y 1 ( t ) and y 2 ( t ) are solutions of (3) and c 1 and c 2 are arbitrary constants, then so is c 1 y 1 ( t ) + c 2 y 2 ( t ) a solution of (3). Date : January 13, 2011. 1 2 KIAM HEONG KWA A linear equation such as (3) is said to be homogeneous if the non- homogeneous term g ( t ) is identically zero. Otherwise, the equation is said to be nonhomogeneous . Remark 1. At times, (3) is also expressed in the form P ( t ) y 00 + Q ( t ) y + R ( t ) y = G ( t ) for some functions P ( t ) , Q ( t ) , R ( t ) , and G ( t ) , where P ( t ) is not iden- tically zero. Second-order linear equations are worth studying because they serve as mathematical models of some important physical processes such as mechanical and electrical vibrations. As an instance, consider a mass m attached to an elastic spring of length l , which is suspended from a rigid horizontal support. An elastic spring has the property that if it is stretched or compressed a distance l which is small compared to its natural length l , then it will exert a restoring force k l proportional...
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