3.1: Differential Equations

Learning Objectives

• Identify patterns in derivatives of exponential functions to match them to first- and second-order differential equations.
• Identify patterns in derivatives of sine and cosine functions to match them to second-order differential equations.
• Derive equations for charge and current in LC circuits as functions of time.

Model

Explore

1. Identify the following for each function:
   • The variable (symbol) that the function shows explicit dependence on.
   • What that symbol represents.
   • The coefficient that appears in front of that variable in the equation.

   Function	Variable Symbol	Variable Represents	Coefficient Symbol
   \(T(t)\)
   \(R(x)\)

2. Describe the structure of each equation, including what is similar and different about the A and B terms.
3. Complete the model by doing the following:
   • Take the derivative of each function and write it on line (1) of the model.
   • Assume \(B=0\) and \(A\neq0\). Rewrite the derivative using the original function (\(T\) or \(R\)). Put this on line (2).
   • Assume \(A=0\) and \(B\neq0\). Rewrite the derivative using the original function (\(T\) or \(R\)). Put this on line (3).
   • Assume \(A\neq0\) and \(B\neq0\). Take the second derivative of each function. Put this on line (4).
   • Substitute the original function into the second derivative. Put this on line (5).

Critical Thinking & Concept Invention

1. What function \(T(t)\) satisfies \(\dfrac{dT}{dt}=\omega T\)? Explain why this works based on derivative rules for exponential functions.
2. What function \(T(t)\) satisfies \(\dfrac{dT}{dt}=-\omega T\)? Explain what is necessary to obtain the negative sign.
3. What function \(T(t)\) satisfies \(\dfrac{d^{2}T}{dt^{2}}=\omega^{2}T\)? Explain why this works based on derivative rules for exponential functions.
4. It is possible to obtain a similar exponential function satisfying \(\dfrac{d^{2}T}{dt^{2}}=-\omega^{2}T\). What might that function be? Consider what determines the coefficient in the second derivative.

Summary

First-order differential equations such as \(\dfrac{d}{dt}\) and \(\dfrac{d}{dx}\) can often be solved analytically by separating variables and integrating. Exponential functions satisfy:

\(\dfrac{dT}{dt}=(\pm)\omega T\)

or

\(\dfrac{dR}{dx}=(\pm)kR\)

The general solutions are:

\(T(t)=Ae^{+\omega t}+Be^{-\omega t}\)

\(R(x)=Ae^{+kx}+Be^{-kx}\)

The sign determines whether \(A\) or \(B\) equals zero. Constants are determined using boundary conditions. For time-based functions, \(\omega\) is often related to the time constant:

\(\tau=\dfrac{1}{\omega}\)

or

\(\omega=\dfrac{1}{\tau}\)

Apply

1. A resistor, voltage source, and inductor are connected in series. The voltage source has an emf of 9 V, the resistor has a resistance of 150 Ω, and the inductor has an inductance of \(30\times10^{-3}\) H. After the current reaches its maximum value, the switch is changed so only the resistor and inductor remain connected.
   
   Recall:
   \(I(t)=\dfrac{\varepsilon}{R}\left(e^{-t/\tau_L}\right)\)
   • Find the current at \(t=0\).
   • Determine constants \(A\) and \(B\) in \(I(t)=Ae^{+\omega t}+Be^{-\omega t}\).
   • Determine \(\omega\).

Model

Explore

1. Describe the structure of each equation, including similarities and differences between the A and B terms.
2. Complete the model:
   • Take the derivative and write it on line (1).
   • Take the second derivative and write it on line (2).
   • Substitute the original function into the second derivative and write it on line (3).

Critical Thinking & Concept Invention

1. Which proportionality aligns with sine and cosine derivatives?
   \(\dfrac{dT}{dt}\propto(\pm)T\)
   or
   \(\dfrac{d^{2}T}{dt^{2}}\propto(\pm)T\)
2. Write the relationship as an equality, including sign and coefficient.
3. Explain why the sign is positive or negative.

Summary

Sine and cosine functions satisfy:

\(\dfrac{d^{2}T}{dt^{2}}=-\omega^{2}T\)

or

\(\dfrac{d^{2}R}{dx^{2}}=-k^{2}R\)

General solutions:

\(T(t)=A\cos(\omega t)+B\sin(\omega t)\)

\(R(x)=A\cos(kx)+B\sin(kx)\)

The period is:

\(T=\dfrac{2\pi}{\omega}\)

Another equivalent form is:

\(T(t)=C\cos(\omega t+\phi)\)

Examples:

• \(\phi=0\): \(C\cos(\omega t)\)
• \(\phi=\dfrac{\pi}{2}\): \(C\sin(\omega t)\)

Apply

1. A circuit has current:
   \(I(t)=(1.2\,A)\sin(6\pi t)\)
   • Find the current at \(t=0\).
   • Determine \(\omega\).
   • Determine the period.
   • Determine the maximum current and the first time it occurs.
   • Determine \(\phi\) if written as \(I(t)=(1.2\,A)\cos(6\pi t+\phi)\).