Exam Seat No:________________
Enrollment No:____________________
C.U.SHAH UNIVERSITY WADHWAN CITY University (Winter) Examination -2013 Subject Name: -Differential Equations
Course Name :M.Sc(Maths) Sem-I Duration :- 3:00 Hours
Marks :70 Date : 27/12/2013
Instructions:(1) Attempt all Questions of both sections in same answer book / Supplementary. (2) Use of Programmable calculator & any other electronic instrument is prohibited. (3) Instructions written on main answer Book are strictly to be obeyed. (4)Draw neat diagrams & figures (If necessary) at right places. (5) Assume suitable & Perfect data if needed.
Q-1
SECTION-I a) Determine the radius of convergence of 𝑒𝑒 𝑥𝑥 . 1
3
(02)
5
b) Evaluate: Γ �2� Γ �2� Γ �2�.
(01)
c) Are sin 𝑥𝑥 and cos 𝑥𝑥 linearly independent? d) Write generating function of Bessel’s function.
(01) (01)
1
Q-2
Q-2
Q-3
Q-3
e) Find 𝑛𝑛 such that ∫−1 𝑃𝑃𝑛𝑛 (𝑥𝑥) 𝑑𝑑𝑑𝑑 = 2 . f) Write Legendre’s equation.
(01)
a) Find the series solution about 𝑥𝑥 = 0 for 𝑦𝑦 ′′ + 𝑦𝑦 = 0. b) Show that 𝑥𝑥 = ∞ is a regular singular point of 𝑥𝑥 2 𝑦𝑦 ′′ + 4𝑥𝑥𝑦𝑦 ′ + 2𝑦𝑦 = 0. c) Solve the differential equation 𝑦𝑦 ′′ + 𝑦𝑦 = sec 𝑥𝑥 by the method of variation parameters. OR a) Find the series solution about 𝑥𝑥 = 0 for 𝑦𝑦 ′ − 2𝑥𝑥𝑥𝑥 = 0. b) Determine the radii of convergence of the following series. 𝑛𝑛! ∞ 𝑛𝑛 2 𝑛𝑛 𝑖𝑖) ∑∞ 𝑛𝑛=1 𝑛𝑛 𝑛𝑛 𝑥𝑥 , (𝑖𝑖𝑖𝑖) ∑𝑛𝑛=0 𝑛𝑛 𝑥𝑥
(05) (05) (04)
a) Find the series solution about 𝑥𝑥 = 0 for 2𝑥𝑥 2 𝑦𝑦 ′′ − 3𝑥𝑥𝑦𝑦 ′ + (3 − 𝑥𝑥)𝑦𝑦 = 0. b) By using Rodrigues formula, find 𝑃𝑃𝑛𝑛 (𝑥𝑥), where 𝑛𝑛 = 0, 1,2,3,4 . OR 0, 𝑚𝑚 ≠ 𝑛𝑛 1 a) Show that ∫−1 𝑃𝑃𝑛𝑛 (𝑥𝑥)𝑃𝑃𝑚𝑚 (𝑥𝑥) 𝑑𝑑𝑑𝑑 = � 2 . Also evaluate , 𝑚𝑚 = 𝑛𝑛 2𝑛𝑛+1
(07) (07)
(01)
c) Determine the singular points of differential equation 2𝑥𝑥 (𝑥𝑥 − 2)2 𝑦𝑦 ′′ + 3𝑥𝑥𝑦𝑦 ′ + (𝑥𝑥 − 2)𝑦𝑦 = 0 and classify them as regular or irregular.
(05) (05) (04)
(07)
1
∫−1 𝑃𝑃3 (𝑥𝑥)𝑃𝑃2 (𝑥𝑥) 𝑑𝑑𝑑𝑑 .
2
b) Prove that: 𝑖𝑖) 𝐽𝐽1 (𝑥𝑥) = �𝜋𝜋 𝑥𝑥 sin 𝑥𝑥, 2
𝑑𝑑
𝑖𝑖𝑖𝑖𝑖𝑖) 𝑑𝑑𝑑𝑑 𝐽𝐽0 (𝑥𝑥) = −𝐽𝐽1 (𝑥𝑥)
2
𝑖𝑖𝑖𝑖) 𝐽𝐽−1 (𝑥𝑥) = �𝜋𝜋 𝑥𝑥 cos 𝑥𝑥, 2
(07)
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Q-4
SECTION-II a) Eliminate the constants 𝑎𝑎 and 𝑏𝑏 from 𝑧𝑧 = (𝑥𝑥 + 𝑎𝑎)(𝑦𝑦 + 𝑏𝑏).
b) Solve: 𝑧𝑧 = 𝑝𝑝𝑝𝑝 + 𝑞𝑞𝑞𝑞 + �1 + 𝑝𝑝2 + 𝑞𝑞 2 . c) Solve: 𝑥𝑥𝑥𝑥 + 𝑦𝑦𝑦𝑦 = 𝑧𝑧. d) Under which condition Pfaffian differential equation in three variables is integrable. e) Write Lagrange’s equation. f) Define: Hypergeometric function. Q-5
Q-5
(01) (01) (05)
a) A necessary and sufficient condition that there exists between two functions 𝑢𝑢(𝑥𝑥, 𝑦𝑦) and 𝑣𝑣(𝑥𝑥, 𝑦𝑦) a relation 𝐹𝐹(𝑢𝑢, 𝑣𝑣) = 0 not involving 𝑥𝑥 or 𝑦𝑦
(07)
c) Eliminate arbitrary function 𝑓𝑓 from 𝑧𝑧 = 𝑓𝑓 � 𝑧𝑧 �. explicitly is that
Q-6
(01) (01)
a) Find integral curve of the simultaneous differential equation 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 = = 2 . 𝑧𝑧𝑧𝑧 −𝑧𝑧𝑧𝑧 𝑦𝑦 − 𝑥𝑥 2 b) Using Picard’s method of successive approximations, find the third 𝑑𝑑𝑑𝑑 approximation of the solution of equation: 𝑑𝑑𝑑𝑑 = 𝑥𝑥 + 𝑦𝑦 2 , where 𝑦𝑦 = 0 when 𝑥𝑥 = 0. c) Eliminate arbitrary function 𝑓𝑓 from 𝑧𝑧 = 𝑥𝑥𝑥𝑥 + 𝑓𝑓(𝑥𝑥 2 + 𝑦𝑦 2 ). OR a) Find integral curve of the simultaneous differential equation 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 = = . 2 2 𝑥𝑥 + 𝑦𝑦 2𝑥𝑥𝑥𝑥 (𝑥𝑥 + 𝑦𝑦)𝑧𝑧 b) Solve (𝑦𝑦 2 + 𝑧𝑧 2 )𝑑𝑑𝑑𝑑 + 𝑥𝑥𝑥𝑥 𝑑𝑑𝑑𝑑 + 𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 = 0 by using Natani’s method. 𝑥𝑥𝑥𝑥
Q-6
(02) (01)
𝜕𝜕(𝑢𝑢 ,𝑣𝑣) 𝜕𝜕(𝑥𝑥,𝑦𝑦)
= 0.
b) Find complete integral of 2(𝑧𝑧 + 𝑥𝑥𝑥𝑥 + 𝑦𝑦𝑦𝑦) = 𝑦𝑦𝑝𝑝2 by using Charpit’s method. OR a) Show that a complete integral of 𝑓𝑓�𝑢𝑢𝑥𝑥 , 𝑢𝑢𝑦𝑦 , 𝑢𝑢𝑧𝑧 � = 0 is 𝑢𝑢 = 𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐𝑐𝑐 + 𝑑𝑑 where 𝑓𝑓(𝑎𝑎, 𝑏𝑏, 𝑐𝑐) = 0. Also find the complete integral of 𝑢𝑢𝑥𝑥 + 𝑢𝑢𝑦𝑦 + 𝑢𝑢𝑧𝑧 − 𝑢𝑢𝑥𝑥 𝑢𝑢𝑦𝑦 𝑢𝑢𝑧𝑧 = 0. b) Prove: (𝑖𝑖) 𝐹𝐹(−𝑛𝑛, 1; 1; −𝑥𝑥) = (1 + 𝑥𝑥)𝑛𝑛 , 𝑎𝑎𝑎𝑎 (𝑖𝑖𝑖𝑖) 𝐹𝐹 ′ (𝑎𝑎, 𝑏𝑏; 𝑐𝑐; 𝑥𝑥) = 𝐹𝐹(𝑎𝑎 + 1, 𝑏𝑏 + 1; 𝑐𝑐 + 1 ; 𝑥𝑥). 𝑐𝑐
(05)
(04) (05)
(05) (04)
(07)
(07)
(07)
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