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You don’t need to solve every questions. Please solve #1,2,5 first and then after that solve as many as you can from backword for 1 hour. I need asap so once you solve one problem please send me thank you.

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Math 108 – Exam # 1 – Fall 2020

Rules

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You may use your textbook and your class notes during the exam.

You need to follow the notation used in our textbook and in our class notes.

You may not use the internet, a smart device (cellphone, watch, etc.), any apps or other

resources to find solutions. Using any of those resources will result in a 0 on the exam

and a report of cheating will be filed with the college.

You are not allowed to collaborate with anyone during the exam.

You must show all work, justify your answers, and write legibly to receive full credit.

A correct answer with unsupported work is an automatic zero on the problem.

You may be selected after the exam to have a private Zoom session with your professor

to explain your work on a problem.

1. Find a general explicit solution to the differential equation.

𝑡

𝑑𝜃

= 3𝜃

𝑑𝑡

2. Consider the initial value problem 𝑦 ′ (𝑡) + 𝑡𝑦 = 𝑦, 𝑦(1) = −2.

a) Determine the independent and dependent variables.

b) Solve the initial value problem using the method of separation of variables.

3. Determine the constant k so that the equation is exact. Then solve the resulting

equation.

(2𝑥 − 𝑦𝑠𝑖𝑛 𝑥𝑦 + 𝑘𝑦 4 )𝑑𝑥 − (20𝑥𝑦 3 + 𝑥𝑠𝑖𝑛 𝑥𝑦)𝑑𝑦 = 0

4. Determine whether the Existence and Uniqueness Theorem (Theorem 1 in our

textbook) implies that the given initial value problem has a unique solution.

√𝑦

𝑑𝑦

= √𝑥

𝑑𝑥

, 𝑦(1) = 1

5. Consider the initial value problem 𝑦

𝑑𝑥

𝑑𝑦

2

+ (1 + 𝑦)𝑥 = 𝑒 −𝑦 sin 2𝑦, 𝑥(𝜋) = 𝑒 𝜋

a) Determine the independent and dependent variables, and whether the equation is

linear or nonlinear.

b) Solve the initial value problem.

6. A tank initially contains 100 liters of pure water. A mixture containing a concentration of

8 grams/liter of salt enters the tank at a rate of 3 liters/min. and the well-stirred mixture

leaves the tank at the same rate. Let A(t) be the amount of salt in the tank. Determine the

amount of salt in the tank at any time, t. How much salt is present after 1 hour?

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Tags:

differential equation

separation of variables

Initial value problem

explicit solution

Uniqueness Theorem

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