MA 345-04 Differential EquationsText : “Fundamentals of Differential Equations”
by Nagle, Saff and Snider
Instructor : Dr. E. Jacobs
Office : COAS 301-36
Office Hours This Week:Wed. and Fri. 3:15 - 4:15 PM
e-mail: [email protected]
Grading System
Exam Average 85%Homework Average 15%
There will be four exams, all equally weighted.Exam 4 will be the final exam.
Overall Average Grade in Course
90 - 100 A80 - 89 B70 - 79 C60 - 69 D0 - 59 F
Some important dates:
Exam 1 Tuesday, September 21Final Exam Tuesday, December 14
Assignment 1 Friday, September 3
Rules for Exams
1. You may not have formulas or notes with you onexams.
2. Put away cell phones and smart watches duringexams.
3. Make-up exams will only be given for in veryspecial circumstances. Arrangements for a make-upexam must be made within 24 hours of the originalexam.
Homework1. Homework must be neat. Show work.2. Upload to Canvas in .pdf format3. Homework must be handed in on time.
Differential Equation
A differential equation is an equation that involvesthe derivative of an unknown function.
dy
dx− 2y = 0
dy
dx+ y = 0
dy
dx+ y = ex
A differential equation could involve partial deriva-tives
∂u
∂t= α2 ∂
2u
∂x2where u = u(x, t)
ih̄∂Ψ
∂t= − h̄2
2m
∂2Ψ
∂x2+ VΨ where Ψ = Ψ(x, t)
Malthusian Model of Growth
Population grows at a rate proportional to the sizeof the population at any point in time.
Malthusian Model of Growth
Population grows at a rate proportional to the sizeof the population at any point in time.
dPdt is proportional to P at any t
Malthusian Model of Growth
Population grows at a rate proportional to the sizeof the population at any point in time.
dPdt is proportional to P at any t
dP
dt= kP
Let A = A(t) be the amount of money in a bankaccount after t years. The bank pays 1 percent in-terest. Find a formula for A(t).
Let A = A(t) be the amount of money in a bankaccount after t years. The bank pays 1 percent in-terest. Find a formula for A(t).
Interest = (Principal)(rate)(time)
Let A = A(t) be the amount of money in a bankaccount after t years. The bank pays 1 percent in-terest. Find a formula for A(t).
Interest = (Principal)(rate)(time)
dA = A · 0.01 · dt
Let A = A(t) be the amount of money in a bankaccount after t years. The bank pays 1 percent in-terest. Find a formula for A(t).
Interest = (Principal)(rate)(time)
dA = A · (0.01) · dt
dA
dt= 0.01A
The rate at which the mass decreases is proportionalto the mass at any given time.
dM
dt= λM (λ is a negative number)
dy
dx= −2x where y(0) = 4
y = −x2 + C
If x = 0 then 4 = −02 + C = C
y = −x2 + 4 (Particular Solution)
dy
dx= −2y where y(0) = 1
Integrate both sides.
y =
∫−2y dx = ???
∫−2 sinx dx = 2 cosx+ C∫−2ex dx = −2ex + C∫−2x3 dx = −1
2x4 + C
dy
dx= 2xy − 2xy2 = 2xy(1− y)
1
y(1− y)dy = 2x dx∫
1
y(1− y)dy =
∫2x dx∫ (
1
y+
1
1− y
)dy =
∫2x dx
ln |y| − ln |1− y| = x2 + C
ln |y| − ln |1− y| = x2 + C
ln
∣∣∣∣ y
1− y
∣∣∣∣ = x2 + C∣∣∣∣ y
1− y
∣∣∣∣ = ex2+C = eCex
2
y
1− y= ±eCex
2
y
1− y= aex
2
y
1− y= aex
2
y = aex2
(1− y)
y = aex2
− aex2
y
y + aex2
y = aex2(
1 + aex2)y = aex
2
y =aex
2
1 + aex2