Lesson: Modeling with Second-Order Differential Equations

In this lesson, we will learn how to use second-order differential equations to model situations in real life.

Worksheet: 5 Questions

Q1:

Consider a body of mass 𝑚 attached to a spring with spring constant 𝑘. The differential equation 𝑚𝑦𝑡=𝑘𝑦dd, where 𝑦 is the vertical displacement, can be used to model the dynamics of this system. However, such a differential equation implies that, once the body starts moving, it will oscillate forever. A better model is one that also considers the effects of frictional forces. If we add a frictional force that is proportional to the velocity of the body, we get the following differential equation 𝑚𝑦𝑡=𝑘𝑦𝑠𝑦𝑡,dddd where 𝑠>0.

Letting 𝑤=𝑘𝑚 and 2𝑏=𝑠𝑚, there are three possible behaviours of the solution to this equation which we can define in terms of 𝑏 and 𝑤 as follows: over damped when 𝑏>𝑤; critically damped when 𝑏=𝑤; and underdamped or oscillatory when 𝑏<𝑤. Which of the following solutions can be used to describe one of these behaviours:

  1. 𝑦 = ( 𝑐 + 𝑐 𝑡 ) 𝑒
  2. 𝑦 = 𝑐 𝑒 + 𝑐 𝑒 , where 𝜆=𝑏+𝑏𝑤 and 𝜇=𝑏𝑏𝑤
  3. 𝑦 = 𝑐 𝑒 ( 𝛽 𝑡 + 𝛾 ) s i n where 𝛽=𝑤𝑏

Q2:

The one-dimensional time-independent Schrodinger equation is given as dd𝜓𝑥=2𝑚[𝑈(𝑥)𝐸]𝜓,

where 𝜓 is a wave function which describes the displacement 𝑥 of a single particle of mass 𝑚, 𝐸 is the total energy, 𝑈 is the potential energy, and is a known constant. Since 𝑈(𝑥)=0 for the particle-in-a-box model, where 0𝑥𝑎, this second-order differential equation becomes 𝜓=𝛼𝜓,𝛼=2𝑚𝐸.where

Find the general solution for this differential equation.

Q3:

Consider a mass 𝑚 that oscillates at the end of a spring having a spring constant 𝑘. The following second-order differential equation describes the vertical displacement 𝑦 of this spring-mass system: 𝑚𝑦𝑥=𝑘𝑦.dd This differential equation neglects the influence of air resistance or frictional forces. Find the general solution which describes the vertical displacement of this spring as a function of time 𝑡.

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