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Unformatted text preview: 3.1/3.4/3.5: HOMOGENEOUS EQUATIONS WITH CONSTANT COEFFICIENTS KIAM HEONG KWA A general secondorder 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. Secondorder equations arise very often in applications. The most wellknown secondorder 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 secondorder 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 secondorder 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. Secondorder 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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 Winter '11
 Kwa
 Differential Equations, Equations, Partial Differential Equations, Vector Space, Mass, Elementary algebra, KIAM HEONG KWA

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