MAT3237 Final Exam Information
Date & Time: 12/08 1:00-3:00pm
Highlights of Important Topics
Recipes for First Order D.E.
SeparableIntegrating Factor
is exact if
/ There is a functionsuch that .
Solve for; the solution is
is an Homogeneous Equations if / Use
Bernoulli’s Equation / Use for
Linear Polynomial Reduction / Use
Reduction of Order Let be a solution of, we can find a second linearly independent solution by solving for .
Second Order D.E. with Constant Coefficients
Consider the roots of the auxiliary /characteristic equation: .
Two distinct real roots ,One real root
Complex roots
Second Order Nonhomogeneous Linear Equations
or
Suppose is a particular solution and is the solution of the complementary equation . Then general solution is .
Findby : Undetermined Coefficients
TryPolynomial of degree n / Polynomial of degree n
Sum / Sum
Product / Product*
Findby : Variation of Parameters
Suppose . Let .
where
Nonlinear equations Assume
Case 1: Dependent variable is missing – use reduction of order .
Case 2: Independent variable is missing – use reduction of order and chain rule.
Laplace Transform
Translation
Translation
Convolution / Product
Derivatives
Integrals
Periodic
Functions
Please pay attention to your presentation. It is very important that you explain your solutions carefully and logically with the systematic methods developed in the lectures. Most points are given to the process of getting the answer. Therefore, please present as much details as possible.
The table on the right will be printed on the exam paper.
Series Solutions of D.E.
Let, then and
The goal is to solve for most of the coefficients.
Expectations
You need to pay attention to the precise arguments of the solutions.
Example 1 Reduction of Order
Let ;;If is a solution then
The transitional statement “If is a solution then” should be in place to explain why
.
Example 2 Exact D.E.
Since , the D.E. is exact.The reason must be given in order to show that the D.E. is exact.
Likewise, the statement “Since the D.E. is exact, there exists such that” is also essential.
Since the D.E. is exact, there exists such thatand
Example 3 General Solutions of
Consider the complementary D.E.: .Consider the auxiliary equation:
In order to solve the nonhomogeneous D.E., we need to solve the complementary D.E.
In order to solve the complementary D.E., we need to solve the auxiliary equation.
So it is important to indicate these reasons with the statements
“Consider the complementary D.E.: .
Consider the auxiliary equation:”
Example 4 Variation of Parameters -must declare .
Let, , , and.Example 5 Partial Fractions – must be done on the right hand side. Must write out the final partial fractions before copy-and-paste to the left hand side.
Please review other important expectations from handouts and PPTs.
Practice Problems
(Disclaimer: This practice exam has no direct relations with the real exam. You need to understand that the problems in the real exam may not resemble the homework problems or the problems in this practice exam. )
1. Use the convolution theorem to find .
2. Solve
3. Find the Laplace transform of the following periodic function.
4. Solve with by eliminating X first.
5. Use power series to solve the D.E.
.
6. Solve .
7. Solve .
8. Solve .
9. Find a particular solution of .
10.Find a particular solution of .
11. Find .
12. Suppose y is a function in x. Solve
with .
You may give implicit solutions.
13. A tank in the form of a right circular cylinder standing on end is leaking water through a circular hole in its bottom. When friction and contraction of water at the hole are ignored, the height h of water in the tank is described by
where and are the cross-sectional areas of the water and the hole, respectively.
(a) Solve for if the initial height of the water is H. Use g = 32.
(b) Suppose the tank is 20 ft high and has radius 2 feet and the circular hole has radius 1 in. If the tank is initially full, how long will it take to empty?
Answers
1.
2.
3.
4.
5. 6.
7.
8. 9.
10.
11.
12.
13. (a) (b)
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