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Separable Equations, Torricelli’s Law | Differential Equations - Exam 2 | MATH 3411, Exams of Differential Equations

Material Type: Exam; Professor: Hollis; Class: DIFFERENTIAL EQUATIONS; Subject: Mathematics; University: Armstrong Atlantic State University; Term: Spring 2009;

Typology: Exams

Pre 2010

Uploaded on 08/04/2009

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Math 3411–Spring 2009
Some review items for Exam 2
THE DATE FOR EXAM 2 IS TUESDAY, MARCH 3
The test will be given in class. You will have the entire class period to work on it.
It is very important that you write neatly and present your answers logically. This helps me a great
deal when determining partial credit. Messy and unreadable work will not be graded.
Questions will cover material from sections 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 4.1, 4.2.
Calculators are allowed, but no books or notes are allowed. You may bring blank scratch paper if you
want.
See the syllabus for an explanation of how the tests count toward your final grade.
Note: There might be sample problems posted on Dr. Hollis’ webpage. Check here:
http://www.math.armstrong.edu/faculty/hollis/classes/DE/
1. Section 3.1 Separable Equations
Be able to use separation of variables, when applicable.
Be able to use the substitution z=y/x when applicable.
2. Section 3.2 Toricelli’s Law
You should memorize that Toricelli’s Law is
a(y)dy
dt =ρy
where a(y) is the cross sectional area of the tank at height y, and ρis a constant.
You’ll need to be able to answer questions about tanks of varying shapes, and
questions about emptying time.
3. Section 3.3 Bernoulli and Riccati Equations
You should be able to solve Bernoulli equations by making the substitution y=um
and finding an appropriate value for mthat renders the problem solvable.
You don’t need to study Riccati equations.
4. Section 3.4 The Logistic Population Model
You should be able to recognize (1) on page 63 as the differential equation that
describes logistic growth, and that (2) is its solution. It’s a good idea to be able to
solve (1); consider trying it as a Bernoulli equation.
Harvesting: You don’t have to memorize (3) on page 65 or the formulas that follow
it, but you should know how to use them to answer questions similar to those in
the section.
5. Section 3.5 Direction Fields
Given a differential equation in the form y0=f(t, y) you should be able to draw a
direction field, find isoclines, and draw sample solutions; see examples 3 and 4 on
page 69.
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Math 3411–Spring 2009

Some review items for Exam 2

  • THE DATE FOR EXAM 2 IS TUESDAY, MARCH 3
  • The test will be given in class. You will have the entire class period to work on it.
  • It is very important that you write neatly and present your answers logically. This helps me a great deal when determining partial credit. Messy and unreadable work will not be graded.
  • Questions will cover material from sections 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 4.1, 4.2.
  • Calculators are allowed, but no books or notes are allowed. You may bring blank scratch paper if you want.
  • See the syllabus for an explanation of how the tests count toward your final grade.
  • Note: There might be sample problems posted on Dr. Hollis’ webpage. Check here:

http://www.math.armstrong.edu/faculty/hollis/classes/DE/

  1. Section 3.1 Separable Equations
    • Be able to use separation of variables, when applicable.
    • Be able to use the substitution z = y/x when applicable.
  2. Section 3.2 Toricelli’s Law
    • You should memorize that Toricelli’s Law is

a(y)

dy dt

= −ρ

y

where a(y) is the cross sectional area of the tank at height y, and ρ is a constant. You’ll need to be able to answer questions about tanks of varying shapes, and questions about emptying time.

  1. Section 3.3 Bernoulli and Riccati Equations
    • You should be able to solve Bernoulli equations by making the substitution y = um and finding an appropriate value for m that renders the problem solvable.
    • You don’t need to study Riccati equations.
  2. Section 3.4 The Logistic Population Model
    • You should be able to recognize (1) on page 63 as the differential equation that describes logistic growth, and that (2) is its solution. It’s a good idea to be able to solve (1); consider trying it as a Bernoulli equation.
    • Harvesting: You don’t have to memorize (3) on page 65 or the formulas that follow it, but you should know how to use them to answer questions similar to those in the section.
  3. Section 3.5 Direction Fields
    • Given a differential equation in the form y′^ = f (t, y) you should be able to draw a direction field, find isoclines, and draw sample solutions; see examples 3 and 4 on page 69.
  1. Section 3.6 Numerical Approximation
    • You should be able to use Euler’s method in the form

yn+1 = yn + hf (tn, yn) wheretn = t 0 + nh

to find approximations to the solution at the points yi, like in example 1 on page

  • You don’t need to learn the “improved” Euler method.
  1. Section 4.1 Existence
  • You should know the definition of a local solution, and know the statement of Theorem 4.1.1 on page 77. You don’t need to know the proof of the theorem, but you should be able to use it to give an explanation of why a certain initial value problem either does, or does not have a solution.
  1. Section 4.2 Uniqueness and its consequences
  • You should know definitions 1, 2, and 3 on pages 80–81.
  • You should know the statement of Theorem 1 on page 82.
  • You should be able to do a problem like Example 3 on page 83.
  • You should read Corollaries 1, 2, and 3 on pages 83–84, but just focus on Corollary

Notes: Good calculus and algebra skills are essential. Based on the results of the most recent quiz, many of you have a lot of very basic difficulties that need to be properly sorted out. If you hope to do well in this course, and are having problems with basic calculus (and even high school algebra) then you need to take it upon yourselves to get the help that you need. The best way to study is to do a lot of problems.