COLLEGE OF ENGINEERING PUTRAJAYA CAMPUSFINAL EXAMINATION

SEMESTER 2, 2009 / 2010

PROGRAMME : Bachelor of Engineering (Honours)

SUBJECT CODE : MATB143

SUBJECT : Differential Equations

DATE : 22 March 2010

TIME : 2 hours 30 minutes

INSTRUCTIONS TO CANDIDATES:

1. This paper contains EIGHT (8) questions in three (3) pages.

2. A table of Laplace Transform is attached.

3. A table of selected basic integrals is provided.

4. Answer all questions.

5. Write all answers in the answer booklet provided.

6. Write the answer to each question on a new page.

THIS QUESTION PAPER CONSISTS OF 3 PRINTED PAGES INCLUDING THIS COVER PAGE.

Page 1 of 13

Semester 2, 2009/2010

Question 1 [15 marks]

(a) Find an implicit solution of the given initial-value problem.

, [7 marks]

(b) Determine the value of for which the given differential equation is exact.

Then, solve the equation by using that value of .

, [8 marks]

Question 2 [12 marks]

Find the general solution of .

Question 3 [13 marks]

Find the general solution of ,

Question 4 [10 marks]

A spring whose constant is 18 N/m is attached with a 1-kg mass. The entire system is

then submerged in a liquid that imparts a damping force numerically equal to 11 times

the instantaneous velocity. Determine the equation of motion if the 1-kg mass is initially

released from a point 4m below the equilibrium position, with an upward velocity of

.

Page 2 of 13

Semester 2, 2009/2010

Question 5 [10 marks]

Obtain two linearly independent power series solutions about x = 0 for the differential equation,

(Write down ONLY the first four non-zero terms for each series solution)

Question 6 [16 marks]

(a) Evaluate the inverse Laplace transform of the function

[7 marks]

(b) Use the Laplace Transform to solve the given integral equation.

[9 marks]

Question 7 [10 marks]

Find the Laplace Transform of the function given by

Question 8 [14 marks]

Use the Laplace Transform to solve the initial-value problem

--- END OF QUESTION PAPER ---

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Semester 2, 2009/2010

SOLUTIONS

1(a) ,

Given,

Hence, the implicit solution of the IVP is

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Semester 2, 2009/2010

1(b)

or

2. .

The corresponding auxiliary equation is:

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Semester 2, 2009/2010

,

but

so,

for

for

Let and be the conjugates of and respectively.

so, and

The general solution of the given differential equation is:

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Semester 2, 2009/2010

3. ,

The associated homogeneous Cauchy-Euler equation:

the corresponding auxiliary equation is

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Semester 2, 2009/2010

let

=

let

=

The general solution of the given Cauchy-Euler equation is

4. Since ,

we have

Auxiliary equation

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Semester 2, 2009/2010

Roots

The general solution is

Now,

The initial conditions are

Solve to obtain

The solution to the IVP is

5.

Let .

, and

Substitute into the differential equation;

– – = 0

– – = 0

+ – – – = 0

– + = 0

and for

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Semester 2, 2009/2010

2 linear independent solutions are

,

,

for any choice of and .

6(a) ℒ-1

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=ℒ-1

= ℒ-1 ℒ-1

= ℒ-1 ℒ-1

= ℒ-1 ℒ-1 ℒ-1

=

6(b) ℒ ℒ

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Semester 2, 2009/2010

or

7.

ℒ =ℒ

=ℒ ℒ ℒ

= ℒ ℒ ℒ

= ℒ ℒ ℒ

=

8.

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Semester 2, 2009/2010

So,

Page 13 of 13