Project $2$: Applications of Second$-$Order Linear ODEs

in Real$-$World Phenomena

Objective

This project aims to explore the diverse applications of second$-$order linear ODEs in real$-$

world scenarios. Students will model, solve, and analyze problems related to mechanical

vibrations, electrical circuits, population dynamics, and structural engineering. Through

this project, students will develop problem$-$solving skills, apply mathematical theories,

and interpret results in practical contexts.

Deliverables

$1.$ A written project report detailing all solutions, analyses, and conclusions.

$2.$ A presentation summarizing the findings and their implications in real$-$world con$-$

texts.

Project Components

$1.$ Mechanical Vibrations

Problem Statement: A spring$-$mass system has a mass of m $=5$ kg and a spring

constant k $=20$ N$/$m$.$ The system is subject to an initial displacement of x$(0)=1$ m and

an initial velocity of dx

dt t$=0=0$ m$/$s$.$

$1.$ Derive the governing second$-$order ODE.

$2.$ Solve the ODE analytically.

$3.$ Discuss the system$$s behavior $($oscillatory motion, period, and frequency$).$

Deliverable: Write the solution x$($t$)$ and interpret the results.

$2.$ Electrical Circuits

Problem Statement: A series RLC circuit has L $=0\mathrm{.}5$ H$,$ R $=4,$ C $=0\mathrm{.}01$ F$,$ and

no external voltage source $($E$($t$)=0).$ The initial charge on the capacitor is q$(0)=10$ C

and the initial current is dq

dt t$=0=0$ A$.$

$1.$ Derive the governing second$-$order ODE for the charge q$($t$)$ on the capacitor.

$2.$ Solve the ODE analytically.

$3.$ Discuss the system$$s behavior $($overdamped$,$ underdamped, or critically damped$).$

Deliverable: Write the solution q$($t$)$ and interpret the results.

$1$

$3.$ Population Dynamics

Problem Statement: A population N $($t$)$ is modeled by the equation:

d$2$N

dt$2+2$dN

dt $3$N $=0.$

$1.$ Solve the ODE analytically with initial conditions N $(0)=5$ and dN

dt t$=0=0.$

$2.$ Interpret the behavior of the population over time.

Deliverable: Write the solution N $($t$)$ and discuss its implications.

$4.$ Pendulum Motion

Problem Statement: A pendulum of length l $=2$ m swings with a small initial dis$-$

placement of $(0)=0\mathrm{.}1$ rad and no initial velocity. The motion is governed by:

d$2$

dt$2+$ g

l $=0,$

where g $=9\mathrm{.}8$ m$/$s$2.$

$1.$ Solve the ODE analytically.

$2.$ Determine the period and frequency of the pendulum$$s motion.

Deliverable: Write the solution $($t$)$ and discuss the pendulum$$s motion.

Evaluation Criteria

$$ Mathematical Solution Accuracy $(40\%)$: Correctness in deriving and solving

the ODEs.

$$ Analysis and Interpretation $(30\%)$: Clarity and depth in explaining the phys$-$

ical significance of the results.

$$ Presentation and Report $(20\%)$: Quality of the final report and presentation.

Submission Guidelines

$1.$ Submit the project report as a PDF document.

$2.$ Prepare a $510$ minute summary for the presentation, focusing on key results and

interpretations