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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