Download GTU BE/B.Tech 2019 Summer 1st And 2nd Sem (New And SPFU) MTH002 Ordinary Differential Equation Question Paper

Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Summer 1st And 2nd Sem (New And SPFU) MTH002 Ordinary Differential Equation Previous Question Paper

1
Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ?I &II (SPFU) EXAMINATION ? SUMMER-2019
Subject Code: MTH002 Date: 010/06/2019

Subject Name: Ordinary Differential Equation

Time: 10:30 AM TO 01:00 PM Total Marks: 70

Instructions:

1. Attempt any five questions.

2. Make suitable assumptions wherever necessary.

3. Figures to the right indicate full marks.

Q.1 (a) I. Form the Differential equation from the solution ?? = ?? 1
cos ?? + ?? 2
sin ?? . 03
II. Verity that ?? = ?? ?? (?? cos ?? + ?? sin ?? ) is a solution of ?? " ? 2?? ? + 2?? = 0. 04
(b) I. Solve 2???? ???? + ?? 2
???? = 0. 03
II. Solve the I.V.P. : ?? ?? ?
+ ?? = 0 , ?? (2) = ?2 04

Q.2 (a) Test the exactness and solve
[(?? + 1)?? ?? ? ?? ?? ]???? ? ?? ?? ?? ???? = 0 ; ?? (1) = 0
07
(b) I. Find the Differential equation of the orthogonal trajectory to ?? = ?? ?? 2
03
II. Find the orthogonal trajectories of the family of Cardioids
?? = ?? (1 + cos ?? ) , where a is parameter.
04

Q.3 (a)
Solve the I.V.P. :
????
????
+ ?? = ?? ; ?? (0) = 0
07
(b) Solve ?? ?? ?
= ?? 2
+ ?? 07

Q.4 (a)
Verify that the functions ?? ?
1
2
and ?? 3
2
form a basis of solutions of
4?? 2
?? ??
? 3?? = 0. Also write general solution of the given equation.
07
(b) I. ?? (?? ) = ?? ?? and ?? (?? ) = ?? ??? are linearly independent or dependent? 02
II. If ?? = ?? ?? (?? 1
cos ?? + ?? 2
sin ?? ) find the Wronskian ?? (?? 1
, ?? 2
). 02
III. Find the General Solution of ?? ??+ ?? ?
? 2?? = 0 . 03

Q.5 (a) Solve the I.V.P. ?? ??
+ 4?? = 8 ?? ?2?? + 4?? 2
+ 2 ; ?? (0) = 2, ?? ?
(0) = 2. 07
(b) Using method of undetermined coefficients, solve ?? ??
+ 4?? = 8 ?? 2
. 07

Q.6 (a) Solve ?? ??
? 3?? ?
+ 2?? = ?? ?? 07
(b) Find the roots of the indicial equation to ?? 2
?? ??
+ ?? ?? ?
? (2 ? ?? )?? = 0
by Frobenius method.
07

Q.7 (a) Find the series solution of (?? 2
+ 1)?? ??
+ ?? ?? ?
? ???? = 0 near ?? = 0 . 07
(b) Find the series solution of ?? ??
= 2?? ? in power of ?? . 07

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