Sections 10.1
Introduction to Differential Equations
Section 10.1: Mathematical Modeling: Setting up a
Differential Equation
Applied Calculus, 3/E by Deborah Hughes-Hallet
Copyright 2006 by John Wiley & Sons. All rights reserved.
Review: (Regular) Equations
Example 1. 2𝑥 3 + 3𝑥 2 − 3𝑥 − 2 = 0
Is 𝑥 = 2 a solution?
No, because 2(2)3 +3(2)2 −3 2 − 2 = −4 ≠ 0.
Is 𝑥 = 1 a solution?
Yes, because 2(1)3 +3(1)2 −3(1) − 2 = 0.
What are all solutions?
1
Use algebra to find 𝑥 = 1, − 2 , −2.
Solve[2 x^3 + 3 x^2 - 3 x - 2 == 0, x]
Example 2. cos 𝜋𝑥 = 𝑥 2
Is 𝑥 = 1 a solution?
No, because cos 𝜋 = −1 ≠ 1 = (−1)2 .
What are all solutions?
Plot[{Cos[Pi*x], x^2}, {x, -2, 2}]
FindRoot[Cos[Pi*x] == x^2, {x, 0.4}]
{x -> 0.438431}
𝑥 ≈ ±0.438
Notice that a solutions for a regular equation is a number.
Definition: Differential Equation
A differential equation is an equation in which a derivative of an unknown
function is one of the terms.
1.
2.
y  x 2  1
y  xe
x2
3. xy  y  0
Applied Calculus, 3/E by Deborah Hughes-Hallet
Copyright 2006 by John Wiley & Sons. All rights reserved.
Definition: Solution to a Differential
Equation
A solution to a differential equation is a function such that when it and/or its
derivatives are substituted into the differential equation, the equation represents a
true statement.
1.
2.
y  x  1
2
y  xe
x2
3. xy  y  0
1 3
1. y  x  x  3
3
1 x2
2. y  e  C
2
3. y  2 ln x  3
#1: y= x2 is not a solution
#3: y=1 is another solution
Applied Calculus, 3/E by Deborah Hughes-Hallet
Copyright 2006 by John Wiley & Sons. All rights reserved.
Definition: Initial Value Problem
An initial value problem is a differential equation with an unknown function together
with the value of that function at some point
y  x  1, y 0   3
2
y  x 2  1, y 1  1
𝑦 = 13𝑥 3 − 𝑥 + 3 is a solution to the first initial value problem.
𝑦 = 1 is a solution to the second initial value problem.
Applied Calculus, 3/E by Deborah Hughes-Hallet
Copyright 2006 by John Wiley & Sons. All rights reserved.
Example
Radioactive carbon (carbon - 14) decays at a rate proportional to the amount
of carbon-14 present.
Let 𝑃(𝑡) be the amount of carbon-14 present at time 𝑡.
dP
 kP
dt
Applied Calculus, 3/E by Deborah Hughes-Hallet
Copyright 2006 by John Wiley & Sons. All rights reserved.
Example
A yam is placed inside a 200°F oven. The temperature of the yam increases at
a rate proportional to the difference between the oven temperature and its
temperature.
Let Y(𝑡) be the temperature (°F) of the yam 𝑡 minutes after placed in the oven.
dY
 k 200  Y 
dt
Applied Calculus, 3/E by Deborah Hughes-Hallet
Copyright 2006 by John Wiley & Sons. All rights reserved.
Example
Morphine is administered to a patient intravenously at a rate of 2.5 mg per
hour. About 34.7% of the morphine is metabolized and leaves the body each
hour.
Let M(𝑡) be the amount (mg) of morphine in the body 𝑡 hours after it was
begun to be administered.
dM
 2.5  0.347M
dt
Applied Calculus, 3/E by Deborah Hughes-Hallet
Copyright 2006 by John Wiley & Sons. All rights reserved.
Example
Josh's credit card debt grows at a rate of 13%. Right now he owes $1,347.17
Let d(𝑡) be Josh’s credit card debt 𝑡 years from now.
d   0.13d
d 0   1347.17
Applied Calculus, 3/E by Deborah Hughes-Hallet
Copyright 2006 by John Wiley & Sons. All rights reserved.
Solve
dy
 t 3  2t 2  8t
dt
Integrating both sides with respect to 𝑡 works.
View the solutions graphically.
Solve
dy
 y 3  2y 2  8y
dt
It is less clear what to do symbolically.
Try guessing constant solutions.
Look at solutions graphically.
Direction Field - Slope Field
dy
3
2
 y  2y  8y
dt
Qualitative
dy
 ky, k  0
dt
Can any of the following curves represent a solution to this
differential equation?
Qualitative
Can the curve below be a solution to any of the following
differential equations?
dy
 ty
dt
dy
 t  1y  1
dt
dy  t  1 

dt  y 
2
Finding Solutions
y  y  0
y  y  0
y  y  0
y  y  0