Download SGBAU BSc 2019 Summer 2nd Sem Mathematics Differential Equations Ordinary n Partial Question Paper

Download SGBAU (Sant Gadge Baba Amravati university) BSc 2019 Summer (Bachelor of Science) 2nd Sem Mathematics Differential Equations Ordinary n Partial Previous Question Paper

AW?1655
B.Sc. Part-l (Semester-ll) Examination
MATHEMATICS
(Differential Equations : Ordinary & Partial)
Paper?lll
Time : Three Hours] [Maximum Marks : 60
Note :?(I) Question No. 1 is compulsory. Solve it in ONE attempt only.
(2) Attempt ONE question from each unit.
1. Choose the correct alternative :
(i) The roots of the equation (D2 ? 4D + 13)2y = 0 are : l
(a) distinct and real (b) real and equal
(0) complex and repeated ((1) None of these
(ii) A linear equation of ?rst order is of the form Y' + PY = Q in which ? l
(a) P is function of Y
(b) P and Q are function of X
(c) P is function of X and Q is function of Y
((1) None of these
(iii) The condition for the partial differential equation Rx, y, z, p, q) = 0 and g(x, y, z, p. q) = 0
to be compatible is that : l
(a) 1w + qu + PJw + q.qu = 0 (b) pr + qu + PJZp + q.qu = 0
(c) pr + J? + PJw + q.qu = 0 (d) None of these
62v 62v 82v Lazv
(iv) The DH. 8x2 + By2 + 622 _c2 612 :0 is called : l
(3) Partial differential equation (b) Ordinary differential equation
(c) Total di?erential equation (d) Linear differential equation
(v) An equation of the form Pp + Qq = R where P, Q, R are the functions of X, Y, Z is
called : 1
(a) Lagrange?s equation (b) Jacobi?s equation
(c) Charpit?s equation (d) Clairaut?s equation
(vi) The particular solution of DE W" + PW' + QW = 0 is y = e" iff : l
(a) P+xQ=O (b)1+p+q=0
(c)1?P+Q=O (d)m1+mP+Q=0
(vii)The solution of PDE (D ? mD')z = 0 is : l
(a) L = F()' + mx) (b) 2 = F'(y ? mx)
(c) z = F(e") (d) None of these
YBC-?l5229 I (Contd.)

(viii) The general form of PDE 01' ?rst order is :
(a) F(xv. y, z, p) 0 (b) F(x, y, 2, q) = o
(C) F(x_ y. 7: P. Q) I 0 (d) F(y. z. p? q) = 0
(ix) The complete integral 0" F(x )3) = G(y, q) is :
(a) 7 = [h(x a)dx (b) _I'k(_v a)dy
(c) z = [h(x a)dx + [My a)dy + b (d) None of these
(x) The DE de + Ndy = l) is exact iff :
(a) ax ?6y (b) (7}
fM?? ?M
(c) ax 6y (d) 6y
[TNlT?l
2. (a) Show that the DE. :
(sin x sin y ? x eY)d_\' ?? (e-? + cos x ? cos y)dx
is exact and hence solve it.
(b) Find the orthogonal trajectory of r? : a? cos n0.
3. (p) Solve the DE. :
(I + x2)dy + 2xy d). ; cot x dx.
(q) Solve :
UNIT?ll
4. (a) Solve the DE. (D2 ? 4)) z c?,
(b) Solve the DE. (x31)2 ? 3x1) ~ 5))" i: x3 sin(log x).
5. (p) Solve the DB. (x21)2 ? xi) + 4)y Z cos(log x).
2 .
(q) Solve thc D.E. 14:1-49?: + 4y =63 +sin 2x,
dx" dx
UNlT?III
6. (a) Solve the: system of DE : I)?x ? 2y ? O and Dzy + 2x = O.
2
(b) Solve the DE. )"'-- y =-*: by variation of parameter.
1+L
YBC?15229 2
U1
UI
(Contd.)

7- (p)
(q)
8. (a)
(b)
9. (p)
(q)
10. (a)
(b)
11. (p)
(q)
Solve xzy" + xy' + 10y = 0 by changing the independent variable from
xtoz=logx. 5
Solve the following D.E. by removing the ?rst derivative :
x%(x%?y)?2xg?:+Zy+x2y=0, 5
UNIT?IV
Solve :
dx = dy = dz . 5
X01 - Z) V(2 - X) 2(X - Y)
Find the complete integral of z = pzx + qzy. 5
Find the general solution of PDE xzp + yzq = (x + y)2. 5
Solve the PDE p2 + q2 = k2. 5
UNIT?V
Solve the DE. (D2 + 3DD' + 2D'2)z = x + y. 5
Solve by Charpits method pxy + pq + qy = yz. 5
The PDE z = px + qy is compatible with any equation f(x, y, z, p, q) = 0 where f is
homogeneous in x, y, z. Prove this. 3
Find a real function v of x and y, reducing to zero when y = 0 and satisfying
52V 62V 2 2
?+?2?=?47?(X +y ). 5
YBC~I5229 3 ' 625