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Answer All Questions. 3 Marks Each.: Page 1 of 2

This document is a practice exam for a course in differential equations. It contains 24 multiple choice and full answer questions testing concepts across 6 modules: 1) First order differential equations; 2) Higher order linear equations; 3) Fourier series; 4) Partial differential equations; 5) Wave equation; 6) Heat equation. Students are asked to show working for full questions worth between 5-10 marks each, addressing topics like solving initial/boundary value problems, finding Fourier series representations, and deriving one-dimensional wave and heat equations.

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Aiswarya S
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42 views2 pages

Answer All Questions. 3 Marks Each.: Page 1 of 2

This document is a practice exam for a course in differential equations. It contains 24 multiple choice and full answer questions testing concepts across 6 modules: 1) First order differential equations; 2) Higher order linear equations; 3) Fourier series; 4) Partial differential equations; 5) Wave equation; 6) Heat equation. Students are asked to show working for full questions worth between 5-10 marks each, addressing topics like solving initial/boundary value problems, finding Fourier series representations, and deriving one-dimensional wave and heat equations.

Uploaded by

Aiswarya S
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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A B2A102 Pages: 2

Reg. No.____________ Name:______________________


APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY
SECOND SEMESTER B.TECH DEGREE EXAMINATION, MAY 2017
MA 102: DIFFERENTIAL EQUATIONS
Max. Marks: 100 Duration: 3Hours
PART A
Answer all questions. 3 marks each.
1. Solve the initial value problem − = 0, (0) = 4, (0) = −2
2. Show that , are linearly independent solutions of the differential equation

−5 +6 =0 −∞< < +∞.What is its general solution?

3. Solve -4 +5 -2y =0

4. Find the particular integral of ( + 4 + 1) = 3


5. Find the Fourier series of f(x)=x , − ≤ ≤
6. Obtain the half range cosine series of f(x)= ,0 ≤ ≤
7. Form the partial differential equation from = ( )+ ( )
8. Solve ( − ) + ( − ) = ( − )
9. Write down the important assumption when derive one dimensional wave equation.
10. Solve 3 +2 =0 with u(x,0)=4 by the method of separation of variables.
11. Solve one dimensional heat equation when k> 0
12. Write down the possible solutions of one dimensional heat equation.
PART B
Answer six questions, one full question from each module.
Module I
13. a) Solve the initial value problem − 4 + 13 = 0 with y(0)= -1 , (0) = 2
(6)
b) Solve the boundary value problem − 10 + 25 = 0 , y(0)=1,y(1)=0 (5)
OR
14. a) Show that ( )= and ( )= are solutions of the differential

equation +8 + 16 = 0 . Are they linearly independent? (6)

b) Find the general solution of ( +3 − 4) = 0. (5)


Module II
15. a) Solve ( +8) = cos + (6)
b) Solve + = tan x by the method of variation of parameters. (5)
OR

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A B2A102 Pages: 2

16. a) Solve +2 +2 = (6)

b) Solve ( + 2 − 3) = (5)
Module III
−1 + , − < < 0
17. a) Find the Fourier series of ( )= (6)
1+ , 0< <
, 0< <1
b) Find the half range sine series of f( ) = (5)
2− , 1< < 2
OR
− ,− < <0
18. a) Obtain the Fourier series of ( ) = (6)
/4, 0 < <
b) Find the half range cosine series of ( ) = , 0 < < (5)
Module IV
19. a) Solve ( −2 + ) = + (6)

b) Find the Particular Integral of −7 −6 = sin ( + 2 ) (5)

OR
20. a) Solve ( + −6 ) = (6)
b) Solve ( − ) +( − ) = − (5)
Module V
21. Solve the one dimensional wave equation = with boundary conditions
(0, ) = 0, ( , ) = 0 all and initial conditions ( , 0) = ( ), =
( ). (10)
OR
22. A sting of length 20cm fixed at both ends is displaced from its position of
equilibrium, by each of its points an initial velocity given by
, 0 < ≤ 10
= , x being the distance from one end. Determine the
20 − , 10 ≤ ≤ 20
displacement at any subsequent time. (10)
Module VI
23. Derive one-dimensional heat equation. (10)
OR
24. Find the temperature in a laterally insulated bar of length L whose ends are kept at
temperature 0°C, assuming that the initial temperature is
, 0 < < /2
( )= (10)
− , /2 < <
***

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