SECTION - A
1. If y1 and y2 are two solution of y" +p (x) y' + q(x) y = 0 then the general solution of this given equation y1 & y2 are : ...................
(a) L.I (b) L.D (c) both (d) N.O.T
2. Order and degree of the D.E ey'" - xy" + y = 0 are respectively :...............................
(a) 3, 1 (b) 3, not defined (c) not defined ,1 (d) both not defined
3. If y = log {sin (x + a)} + b where a & b are constant, is the primitive, then the corresponding Lowest order D.E. is :.........
(a) y" = - (1 + (y')2) (b) y" = y2 - (y')2 (c) y" = 1+ (y')2 (d) y" = y' + y2
4. Consider the D.E 2cos (y2) dx - xy sin (y2) dy = 0 has I.F:........................................
(a) ex (b) e-x (c) 3x (d) x3
5. The solution of D.E dy/dx = (y2 cos?x + cos?y)/(x sin?y-2y sin?x)
y (Π/2) = 0 is:...................................................
(a) y2 cos x + x sin y = 0 (b) y2 sin x + x cos y = Π/2 (c) y2 sin x + x sin y = 0 (d) y2 cos x + x cos y = Π/2
6. For the D.E xy' - y = 0 which of the following function is not a I .F:.....................
(a) 1/x2 (b) 1/y2 (c) 1/xy (d) 1/(x+y)
7. If y1 (x) and y2 (x) are solution of y" + xy' + (1 - x2) y = sin x. then which of the following is also a solution:...........................
(a) y1 (x) + y2 (x) (b) y1 (x) - y2 (x) (c) 2y1 (x) - y2 (x) (d) N.O.T
8. The D. E (3a2x2 + by cos x ) dx + (2sin x - 4ay3) dy = 0 is exact for:..............................
(a) a = 3, b = 2 (b) a = 2, b = 3 (c) a = 3, b = 4 (d) a = 2, b = 5
9. The orthogonal trajectory to the family of circles x2 + y2 = 2 cx is define by the D. E.
(a) (x2 + y2) y' = 2xy (b) (x2 - y2) y' = 2xy (c) (y2 - x2) y' = xy (d) (y2 - x2) y' = 2xy
10. Let y = Φ (x) and y = ψ (x) be two solution of y" - 2xy' + (sin x2)y = 0 s. t. Φ (0) = 1, Φ' (0) = 1 & ψ (0) = 1 ψ' (0) = z Then the value of w (x):.................................
(a) cx3 (b) cx3/2 (c) cx2 (d) N.O.T
11. The D. E. dy/dx = K (a - y) (b - y), when solved with the condition y (0) = 0, fields the result:......................................................
(a) (b (a-y))/(a (b-y)) = e(a- b) kx (b) (b (a-x))/(a (b-x)) = e(b- a)kx (c) (a (b-y))/(b (a-y)) = e(a- b)kx (d) N.O.T
12. Which of the following D.E. D2 y + sin (x + y) = sin x:.................................................
(a) Linear, Homogeneous (b) Non linear, homogenous (c) linear, non homogenous (d) N.O.T
13. which of the following is not an I.F of xdy - ydx = 0:...................................................
(a) 1/x2 (b) 1/(x2 + y2 ) (c) 1/xy (d) x/y
14. The value of 1/(xd+1) ex is:...........................
(a) log x (b) log?x/x (c) log?x/x2 (d) N.O.T
15. Let y1 (x) = 1 + x and y2 (x) = ex be two solution of y" (x) + p (x) y' (x) + q(x) y = 0
(a) 1 + x (b) (1+x)/x (c) (-1-x)/x (d) N.O.T
16. The complete solution for the ordinary D. E. is D2y + pDy + qy = 0 is y = c1e-x + c2e-x which of the following is a solution of the D.E D2y + pDy + (q+1) y = 0:....................
(a) e-3x (b) xe-x (c) xe-2x (d) x2e-2x
17. Solution of log dy/dx = 3x + 4y, y (0) = 0
(a) e3x + 3e-4y = 4 (b) 4e3x - e-4y = 3 (c) 3e3x + 4e4y = 7 (d) 4e3x +3e-4y = 7
18. P.I of (D2 - 2D + 1) y = xex sin x is :.........
(a) ex (x cos x + 2 sin x)
(b) -ex (2 cos x + x sin x)
(c) -ex (x cos x + 2 sin x)
(d) N.O.T
19. The orthogonal trajectory of the family of curve an-1y = xn is :.......................................
(a) xn + n2y = constant (b)xy2 + x2 = constant (c) n2x + yn= constant (d) n2x - yn= constant
20. if Φ(x) is differential function,then solution of dy +{y Φ'(x) -Φ(x) Φ'(x)} dx =0 is
(a) y = (Φ(x)-1) + ce-Φ(x) (b) y . Φ(x) = Φ (x)2 + c (c) y . Φ(x) = Φ (x) eΦ(x)+ c (d) N.O.T
21. dy/dx = e-2y and y = 0 when x = 5 then the value of x when y = 3:....................................
(a) e5 (b) e6+1 (c) (e6 + 9)/2 (d) N.O.T
22. The solution of dy/dx = (x2 + y2 + 1)/2xy satisfying y (1) = 1 is:...................................................
(a) A system of circle
(b) A system of hyperbola
(c) A system of ellipse
(d) A system of parabola
23. The orthogonal trajectory of the family if curve y = c1 x3 are:.......................................
(a) circle (b) ellipse (c) parabola (d) N.O.T
24. (x + 2y3) dy/dx = y has solution:...................
(a) circle (b) ellipse (c) hyperbola (d) N.O.T
25. An I.F of sin h y dx + cos h y dx = 0 is :...
(a) ex (b)x (c) y (d) xy
26. The equation whose solution is self orthogonal is:..................................................
(a) p - 1/p = p2 (b) (px + y) (x - yp)- λp = 0 (c) (px + y) (x + yp) - λp = 0 (d) N.O.T
27. Let y (x) be the solution of the I.V.P y3 - y2 + 4y1 - 4y = 0 Y (0) = y' (0) = 2, y" (0) = 0. Then the value of y (Π/2) is :...............
(a) ((4eΠ/2-6))/5 (b) ((6eΠ/2-4))/5 (c) ((8eΠ/2-2))/5 (d) N.O.T
28. Solve (3 - x) y" - (9 - 4x) y' + (6 - 3x)y = 0 then solution is:......................................
(a) e-x (b) e2x (c) e3x (d) N.O.T
29. If y = x is a solution of the D. E. y" - (2/x2 +1/x) (xy'-y) = 0 Then its general solution is :......................................................
(a) (α+βe-2x)x (b) (α + βe2x)x (c) (αx + β)x (d) N.O.T
30. y2 + a2 y = cosec ax then x lim→0 y(x)
(a) 1 (b) 2 (c) does not exists (d) N.O.T
SECTION - B :
1. The I .F. of x.dy/dx + (3x + 1)y = xe-2x is:.....
(a) xe3x (b) x2e3x (c) xe2x (d) e3x
2. Let T1 (x) and T2(x) be the two solution of twice order D. E. then W of the function T1 (x) = x2 & T2(x) = x |x| is L.D. for:...........
(a) x > 0 (b) x < 0 (c) x = 0 (d) N.O.T
3. The D.E 2ψd? - (3 ψ- 2?) d ψ = 0:..........
(a) Exact & homogeneous but not linear
(b) Homogeneous & linear but not exact.
(C) exact linear but not homogenous
(d) N.O.T
4. If the general solution of D.E ay" + by" + cy' + dy = 0 is linear spanned by ex sin x & cos x, then which of the following hold at x = 0:...............................................................
(a) a + b - c - d = 0 (b) a + b + c +d (c) c - a = 0 (d) d - b = 0
5. Let f, g: [ -1, 1]→ R where f (x) = x3, g (x) = x2 |x| then:..............................................
(a) f, g are L. I on [ -1, 1] (b) f, g are L. D on [ -1, 1] (c) f (x) g' (x) - f' (x) is not identically zero on [- 1, 1] (d) f (x) g' (x) - f' (x) g (x) is equal to zero
6. The general solution of D. E 4x2 y" - 8xy' + 9y = 0 is /are:......................................
(a) (c1 + c2 log x) x3/2
(b) (c1 + c2 logx) x-3/2
(c) solution not exits x < 0
(d) solution exist when x ≥ 0
7. Consider the D. E dy/dx - 2x = Φ(x) XER, satisfying y (0) = 0
{0 when x ≤ 0
Φ(x) = this I. V. P
{1 when x > 0)
(a) has a continuous solution which is not different at x = 0
(b) has a continuous solution which is different at x = 0
(c) has a continuous solution which is not different at x = R
(d) N.O.T
8. The D.E (d2ψ)/(d?2 ) + ψ = 0 satisfying y (0) = 1, y (Π) = 0 has solution :....................................
(a) unique (b) more than one (c) no solution (d) N.O.T
9. Consider the I. V. P dy/dx xy3, y (0) = 0 (x, y) ∈ R × R then:..................................................
(a) 3/2 y 2/3 = x2/2 (b) unique solution (c) more than (d) N.O.T
10. The I.V.P y1 = √y , y (0) = α α ≥ 0 has
(a) at least 2 solution if α > 0 (b) no solution α > 0 (c) at least one solution if α > 0 (d) a unique solution if α > 0
SECTION - C
1. Consider the D.E dy/dx = ay - by3, where a, b> 0 & y (0) = 0, As x → ∞ then I .F,..........
2. In D. E. (D3 - 3D2 + 4) y = 0 & constants tends zero the solution is ...........................
3. If g (x, y) dx + (x + y) dy = 0 is an exact different equation and g (x, 0) = x2 then g (1, 2) is..........................................................
4. Consider the D.E. dy/dx - y = - y2 then (n lim→∞ y (x) is equal to ...............................
5. D. E. y" - (y')2 = 0 if y(0) = 1 & y (1) = 2 then y (3) = ?...............................................
6. D. E. y' = 2 x-y if y (0) = 1 then y (2) equal to .....................................................................
7. Given that there is a common solution of the following equation: y' + 2y = ex y2, y (0) = 1Y" - 2y' + αy = 0 Then the value of ∝.............................................................
8. The I. F. (x7y2 + 3y )dx + (3x8 y - x) dy = 0 is xm yn then the value of m...........................
9. The complete solution for the O. D. E (d2y)/dx2+ p.dy/dx + qy = 0 is y = c1e-x + c2 e-3x then the value of P...................................
10. The maximum no. of L. I. solution of the D.E.(d4y)/dx4 = 0 with the condition y (0) = 1, is..................................
11. If y = x cos x is solution of an nth order L. D.E. with real constant coefficient then least possible value of n is...........................
12. Solution of the I. V. P xy' - y = 0 with y (1) = 1 at x = 2 is .............................................
13. If x = y1 - (y13)/l3 + (y15)/l5 .............∀x∈R { y = dy/dx then the order of D. E...................................
14. Let y1 (x) & y2 (x) be the L.I. solution of xy" + 2y' + xexy = 0 if w (1) = 2 then w (5) =?................................................................
15. (x2 - x) dy/dx = (2x - 1) y, y (x0) = y0 has a unique solution if (x0, y0) is...........................
16. If y (x) satisfies dy/dx + 2y = 2 + e-x with y (0) = 0 n lim→ ∞ g (x)equal.............................
17. If an integral curve of the differential equation (y - x) dy/dx = 1 passes through (0, 0) & ( α,1) then α is equal to:............
18. If y (t) is a solution of the D. E. y" + 4y = 2 et then t lim→ ∞ e-t y (t):........................
19. The D. E. y" + 6y' + 9y = 50e2x The P. I. at x = 0
19. Let y be a fn of x satisfying dy/dx = 2x3 √y - 4 xy of y (0) = 0
Then y (1)......................................................
20. Let y be a fn of x satisfying dy/dx = 2x3 √y - 4 xy of y (0) = 0
Then y (1).....................................................