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Maths 205 (Final) BB

The document is a final exam for a mathematics course consisting of 8 questions worth a total of 45 marks. The exam covers various topics in differential equations including: [1] solving differential equations; [2] finding general solutions; [3] using Laplace transforms to solve initial value problems; and [4] using the method of variation of parameters. Students are instructed to show their work clearly and calculators are not allowed.

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Saleh Almadhoob
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99 views9 pages

Maths 205 (Final) BB

The document is a final exam for a mathematics course consisting of 8 questions worth a total of 45 marks. The exam covers various topics in differential equations including: [1] solving differential equations; [2] finding general solutions; [3] using Laplace transforms to solve initial value problems; and [4] using the method of variation of parameters. Students are instructed to show their work clearly and calculators are not allowed.

Uploaded by

Saleh Almadhoob
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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University of Bahrain

College of Science
Department of Mathematics
First Semester 2018/2019
Maths-205 Final Exam

Date: January 10, 2019


Time: 2 hour (11:30-13:30) Max. Marks: 40

Student Name:
Serial No:

Student Number: Section:

Please check that this test consists of 8 questions.


Show your work clearly.
Calculator is not allowed

Questions 1 2 3 4 5 6 7 8 Total
Max. 5 5 5 5 6 6 8 5 45
Marks

Marks
obtained

1
Question 1. solve the differential equation
𝑥 3 𝑦 ′′′ − 𝑥 2 𝑦 ′′ + 2𝑥𝑦 ′ − 2𝑦 = 0 , 𝑥 > 0 [5]

2
−𝑥 2
Question 2. Given that 𝑦1 = − is a solution to 𝑦 ′′ + 2𝑦 ′ − 3𝑦 = 𝑥
3 9
𝑒 2𝑥
and 𝑦2 = is a solution to 𝑦 ′′ + 2𝑦 ′ − 3𝑦 = 𝑒 2𝑥 . Find the
5

general solution to 𝑦 ′′ + 2𝑦 ′ − 3𝑦 = 4𝑥 − 5𝑒 2𝑥
[5]

3
Question 3. Solve the DE
4𝑥𝑦𝑦 ′ = 2𝑦 2 − 𝑥2 [5]

4
Question 4. Find the general solution of differential equation.
𝑦 ′′′′ + 𝑦 ′′′ − 7𝑦 ′′ − 𝑦′ + 6𝑦 = 0 [5]

5
Question 5. Solve Initial Value problem (IVP) using Laplace transform
𝑦" − 2𝑦′ + 5𝑦 = −8𝑒 −𝑡 , 𝑦(0) = 2, 𝑦′ (0) = 12 [6]

6
Question 6. Solve using Laplace transform

𝑥 ′ (𝑡) = 𝑥 − 𝑦, 𝑥(0) = −1 ,
𝑦 ′ (𝑡) = 2𝑥 + 4𝑦, 𝑦(0) = 0 [6]

7
Question 7. Using the method of variation of parameter find the general solution of
𝑥 2 𝑦" + 𝑥𝑦′ − 𝑦 = 𝑥 𝑙𝑛𝑥, ( 𝑥 > 0) [8]

8
Question 8. Let 𝑦1 and 𝑦2 be the solution of the differential equation
𝑦" + 𝑎(𝑥)𝑦 = 0 (1).
Show that 𝑍 = 𝑦1𝑦2 solution for differential equation.
𝑍 ′′′ + 4𝑎(𝑥 )𝑍 ′ + 2𝑎′ (𝑥 )𝑍 = 0 (2)
(Bonus, 5 marks)

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