Lesson: Modeling with Second-Order Differential Equations

Mathematics

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 π‘š attached to a spring with spring constant π‘˜. The differential equation π‘šπ‘¦π‘‘=βˆ’π‘˜π‘¦dd, 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 π‘šπ‘¦π‘‘=βˆ’π‘˜π‘¦βˆ’π‘ π‘¦π‘‘,dddd where 𝑠>0.

Letting 𝑀=π‘˜π‘šοŠ¨ and 2𝑏=π‘ π‘š, there are three possible behaviours of the solution to this equation which we can define in terms of π‘οŠ¨ and π‘€οŠ¨ as follows: over damped when 𝑏>π‘€οŠ¨οŠ¨; critically damped when 𝑏=π‘€οŠ¨οŠ¨; and underdamped or oscillatory when 𝑏<π‘€οŠ¨οŠ¨. Which of the following solutions can be used to describe one of these behaviours:

  1. 𝑦=(𝑐+𝑐𝑑)π‘’οŠ§οŠ¨οŠ±οŒ»ο
  2. 𝑦=𝑐𝑒+π‘π‘’οŠ§οŠ±οΏοοŠ¨οŠ±οŽ€ο, where πœ†=𝑏+βˆšπ‘βˆ’π‘€οŠ¨οŠ¨ and πœ‡=π‘βˆ’βˆšπ‘βˆ’π‘€οŠ¨οŠ¨
  3. 𝑦=𝑐𝑒(𝛽𝑑+𝛾)sin where 𝛽=βˆšπ‘€βˆ’π‘οŠ¨οŠ¨

Q2:

The one-dimensional time-independent Schrodinger equation is given as ddοŠ¨οŠ¨πœ“π‘₯=2π‘šβ„[π‘ˆ(π‘₯)βˆ’πΈ]πœ“,

where πœ“ is a wave function which describes the displacement π‘₯ of a single particle of mass π‘š, 𝐸 is the total energy, π‘ˆ is the potential energy, and ℏ is a known constant. Since π‘ˆ(π‘₯)=0 for the particle-in-a-box model, where 0≀π‘₯β‰€π‘Ž, this second-order differential equation becomes πœ“β€²β€²=π›Όπœ“,𝛼=βˆ’2π‘šπΈβ„.where

Find the general solution for this differential equation.

Q3:

Consider a mass π‘š that oscillates at the end of a spring having a spring constant π‘˜. The following second-order differential equation describes the vertical displacement 𝑦 of this spring-mass system: π‘šπ‘¦π‘₯=βˆ’π‘˜π‘¦.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 𝑑.

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