# Worksheet: Modeling with Second-Order Differential Equations

In this worksheet, we will practice using second-order differential equations to model situations in real life.

**Q1: **

Consider a body of mass attached to a spring with spring constant . The differential equation , where 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 where .

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

- , where and
- where

- Ab
- Ba
- CEach solution describes one of the different behaviours
- Dc

**Q2: **

The one-dimensional time-independent Schrodinger equation is given as

where is a wave function which describes the displacement of a single particle of mass , is the total energy, is the potential energy, and is a known constant. Since for the particle-in-a-box model, where , this second-order differential equation becomes

Find the general solution for this differential equation.

- A
- B
- C
- D

**Q3: **

Consider a mass that oscillates at the end of a spring having a spring constant . The following second-order differential equation describes the vertical displacement of this spring-mass system: 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 .

- A,
- B,
- C,
- D,

**Q4: **

Consider a mass that oscillates at the end of a spring having a spring constant . The following second-order differential equation describes the vertical displacement of this spring-mass system:

Such a differential equation implies that mass , once started, will simply oscillate up and down forever! This differential equation neglects the influence of frictional forces. Let us assume that there is a retarding force which is proportional to the velocity of motion where is treated as a proportionality constant . The aforementioned differential equation now becomes

Let and , where .

There are three possible types of solutions which depend upon the relative size of and , including the following:

overdamped if ,

critically damped if ,

underdamped or oscillatory if .

Find the general solution for this differential equation, which describes the vertical displacement of this spring as a function of time for โoverdampedโ motion.

- A, ,
- B, ,
- C, ,
- D, ,

**Q5: **

Consider a mass that oscillates at the end of a spring having a spring constant . The second-order differential equation describes the vertical displacement of this springโmass system.

Such a differential equation implies that mass , once started, will simply oscillate up and down forever. This differential equation neglects the influence of frictional forces. Let us assume that there is a retarding force that is proportional to the velocity of motion where is treated as a proportionality constant . The aforementioned differential equation now becomes

Let and where .

There are three possible types of solutions that depend on the relative size of and , including the following:

overdamped if ,

critically damped if ,

underdamped or oscillatory if .

Find the general solution for this differential equation, which describes the vertical displacement of this spring as a function of time for the โcritically dampedโ motion."

- A
- B
- C
- D