SECTION - A
1. If curve cuts every member of a given family of curve at an angle θ (≠90°), then it is called:...........................................................
(a) trajectory (b) orthogonal trajectories (c) oblique trajectories (d) N.O.T
2. The orthogonal trajectories of the circle x2 + y2 = a2 is given by:.........................................
(a) a circle (b) a parabola (c) an ellipse (d)a straight line u = mx
3. The particular integral of nth order differential equation contains:.........................
(a) (n + 1) arbitrary constants (b) n arbitrary constants (c) one arbitrary constants (d) N.O.T
4. The complete primitive can be obtained by:....................................................................
(a) C. F. (b) P.I. (c) C. F. + P. I. (d) N.O.T
5. If y = emx is a solution of linear equation of second order then:.........................................
(a) m2 + Pm + Q = 0 (b) m2 + Pm + Q = 1 (c) m2 + Pm + Q ≠ 0 (d) N.O.T
6. The order of the differential equation of all parabolas whose axis of symmetry is along x-axis is of order:...................................
(a) 2 (b) 3 (c) 1 (d) N.O.T
7. Education of the curve passing through (3, 9) and which satisfies the differential equation dy/(dx) = x + 1/x2 is:.....................................
(a) 6xy = 3x2- 6x + 29 (b) 6xy = 3x3 - 29x + 6 (c) 6xy = 3x3+ 29x - 6 (d) N.O.T
8. The slope of the tangent at (x, y) to a curve passing thro' (1, Π/4) is given by y/x - cos2 y/x, then the equation of the curve is :..................
(a) y = tan-1 [log(e/x) ]
(b) y = x tan-1 [log(x/e) ]
(c) y = tan-1 [log(e/x) ]
(d) N.O.T
9. The solution of differential equation
yy' = x [y2/x2 +(Φ(y2/x2 ))/(Φ'(y2/x2 ) )] is:....................................
(a) Φ(y2/x2 ) = cx2 (b) x2Φ(y2/x2 ) =c2y2 (c) x2Φ(y2/x2 ) = c (d) Φ(y2/x2 ) = cy/x
10. The transformation which transform the Bernoulli's equation to a linear differential equation is :.......................................................
(a) yn+1 = v (b) yn-1 = v
(c) y-n+1 = v (d) y-n-1 = v
11. If V is a function of x, then the value of 1/(f(D))xV is :.............................................................
(a) x.1/(f(D)).V + (f' (D))/(f (D)) V (b) V 1/(f(D)) V + (f^' (D))/(f (D)) V (c) x 1/(f(D)) V - (f^' (D))/({f(D) }2 ) V
(d) V.1/(f(D)).V + (f' (D))/({f(D) } 2 ) V
12. Complementary function of D4 + 2D2 + 1) y = ex is:..............................................................
(a) c1 cos x + c2 sin x + c3 ex + c4 e-x
(b) (c1 + c2x) cos x + (c3 + c4 x)sin x
(c) (c1 + c2x) e-x + (c3 + c4 x) ex
(d) (c1 + c2x) ex + c3 e2x + c4 e-3x
13.The orthogonal trajectories of family of parabolas y2 = 4a (x+ a) is:
(a) x2 + y2 = c2 (b) x2 + 2y2 = c2 (c) y2 = 4c (x + c) (d) x2 = 4c (x + a)
14. If a curve cuts every member of a given family of the curve at an angle θ (≠90°), then it is called:.........
(a) trajectory (b) orthogonal trajectories
(c) oblique trajectories (d) N.O.T
15. The orthogonal trajectories of the circle x2 + y2 = a2 is given by:........................................
(a) a circle (b) a parabola (c) an ellipse (d) a straight line y = mx
16. If y = emx is a solution of linear equation of second order then:
(a) m2 + Pm +Q = 0 (b) m2 + Pm +Q = 1 (c) m2 + Pm +Q ≠ 0 (d) N.O.T
17. The curve for which subnormal is a constant, is :.........................................................
(a) a circle (b) a parabola (c) an ellipse (d) N.O.T
18. The function f (θ) = d/dθθ∫θdx/(1-cosθ cos x) satisfies the differential equation:.............
(a) df/dθ+ 2f (θ) cot θ= 0
(b) df/dθ- 2f (θ) cot θ= 0
(c) df/dθ+ 2f (θ) = 0 (d) N.O.T
19. The order of the differential equation whose general solution is given by
Y = (c1 + c2) cos (x + c3) - c4 ex+c5, where c1, c2, c3, c4, c5 are arbitrary constants is:
(a) 5 (b) 4 (c) 3 (d) 2 (e) N.O.T
20. If x dy/dx + y = x. (Φ(xy))/(Φ^(xy)), then Φ (xy) is equal to (where k is an arbitrary constant):..........
(a) k ex2/2 (b) key2/2 ) (c) k exy/2 (d) N.O.T
21. Let f (x) be differentiable on the interval (0, ∞) such that f (1) = 1, and t → lim x) t2f(x)-x2 f(t))/(t-x) = 1 for each x > 0. Then f (x) is :....
(a) 1/3x+2/3 x2 (b) -1/3x+4/3 x2 (c) -1/x+2/x2 (d) 1/x
22. The differential equation dy/dx = √( 1-y2 )/y determines a family of circles with:.........
(a) Variable radii and fixed centre at (0,1)
(b) Variable radii and fixed centre at (0,- 1)
(c) fixed radius 1 and variable centre along the x-axis
(d) fixed radius 1 and variable centre along the y-axis
23. Let y' (x) + y (x) g'(x) = g (x) g' (x), y (0) = 0, x ∈ R, where f' (x) denotes d/dx (f(x)) and g (x) is a given non-constant differentiable function on R with g (0) = g (2) = 0. Then the value of y (2) is
(a) 0 (b) 1 (c) -1 (d) 2
24. Let f: [1, ∞) →[2, ∞) be a differentiable function such that f (1) = 2. If 6 ∫_1^x?f (t)dt? = 3x f (x) - x3 For all x≥1, then the value of f (2) is:.........................................................................
(a) 3 (b) 4 (c) 5 (d) 6
25. If y (x) satisfies the differential equation y'-y tan x = 2x sec x and y (0) = 0, then:......................................................................
(a) y (Π/4) = Π2/8√(2), y' (Π/3) = 4Π/3 + 2Π2/3√(3)
(b) y (Π/4) = Π2/4√(2), y' (Π/4) = Π2/18
(c) y (Π/3) = Π2/(9), y' (Π/3) = 4Π/3 + Π2/3√(3)
(d y (Π/3) = Π2/4√(2), y' (Π/3) = Π2/18
26. Assume that all the zeros of the polynomial an xn + an-1 xn-1+.....+ a1x + a0 have negative real parts. If u (t) is any solution to the ordinary differential equation. an (dnu)/dtn + an-1 (dn-1u)/dtn-1 +.....+a1 (du )/dt + a0u= 0, then limt→∞ u (t) is equal to :............
(a) 0 (b) 1 (c) n - 1 (d) ∞
27. The product W (y1, y2) P (x) equals:
(a) y2, y1" - y1, y2" (b) y1, y2' - y2, y1"
(c) y1'y2" - y2'y1" (d) y2'y1' - y1"y2"
28. If y1 = e2x and y2 = xe2x, then the value of P (0) is :...............................................................
(a)4 (b) -4 (c) 2 (d) -2
29. If a transformation y = uv transforms the given differential equation
f(x) y" - 4f' (x) y' + g (x) y = 0 into the equation of the form v" + h (x) v = 0, then u must be :.............
(a) 1/f2 (b) xf (c) 1/2f (d) f2
30. The initial value problem x dy/dx = y + x2, x > 0; y (0) = 0 has:.....................................................
(a) infinitely many solutions
(b) exactly two solutions
(c) a unique solutions
(d) no solution
SECTION - B
1. For a D. E. (Hx2) y'+ 2xy = 4x2 is:....................
(a) Has integrating factor 1+x2
(b) a linear Homogeneous D. E.of order 1.
(C) has I. F 1/2 log (1 + x2) (d) N.O.T
2. 1/N (∂M/∂x-∂N/∂y) = h (y) Then the I. F. of D. E. Ndn + Mdy is :......................................................
(a) eft(x)dx (b) ef-t(x)dx (c) ef(x) (d) N.O.T
3. (D2 - 6D + 7) = ex + e-x is a D. E. then:............
(a) P.I = ex/2 + ex/14 (b) P.I = ex/3 + e(-x)/14 (c) root of equation 2+ √3 I (d) N.O.T
4. Let y1(x) and y2(x) form fundamental set of solutions to the differential equation (d2y)/dx2) + p (x) dy/dx + q (x) y = 0, 0 ≤ x ≤ b,
Where p (x) and q (x) are continuous in [a, b] and x0 is a point in (a, b). Then.
(a) both y1(x) and y2(x) cannot have a local maximum at x0
(b) both y1(x) and y2(x) cannot have a local minimum at x0
(c) y1(x) cannot have a local maximum at x0 and y2 (x) cannot have local minimum at x0 simultaneously.
(d) both y1(x) and y2(x) cannot vanish at x0 simultaneously.
5. Let D2y - q (x) = 0, 0 ≤ x < ∞, y (0) = 1, D (y) (0) = 1, where q (x) is monotonically increasing function. then
(a) y (x) →∞ as x →∞
(b) D (y) →∞ as x →∞
(c) y (x) has finitely many zero in [0, ∞]
(d) y (x) has infinitely many zero in [0, ∞]
6. A family of curve x2/a2 + y2/ b2λ = 1 then:..............................................................................
(a) O. T is y2 + 2a2 log x + x
(b) O. T is y2 - 2a2 log x + x
(c) self orthogonal
(d) not self orthogonal
7. P. I of D. E. y" + 2y' + 3y = 1 + ex at x →-∞:...............................................................................
(a) 1/3 (1 + e-x) (b) 1/3 + 1/6 ex (c) 1/3 - 1/6 ex (d) N.O.T
8. x2 D2 y + 4x Dy + 2y = ex then P. I:........................
(a) at x →-∞ is 1 (b) at x →∞ is 1 (c) at x →∞ is 0 (d) at x →-∞ is 0
9. In D. E D2 y + 4y = tan 2x (by V. of Parameter):.................................................................
(a) W = 2x (b) W = 2 (c) B = - 1/4 cos 2x (d) A = - 1/4 sin 2x
10. In D. E. D2 y - 2 Dy + y = x ex sin x:......................
(a) W = e2x (b) W = ex
(c) A = - ex (x sin x + 2 cos x)
(d) B = cos x
SECTION - C :
1. x dy/dx = (x2 - x - 1) y + (x- 1), then k is equal to:..................................................................................
2. The orthogonal trajectory of x2/(4+λ) + y2/(9+λ) = 1,
Where λ is parameter is A then the value of λ at (2, 3) in A.....................................................................
3. Let Φ be a differentiable function on [0, 1] satisfying Φ' (x) ≤ 1 + 3 Φ (x) for all x ∈ [0, 1] with Φ (0), then Φ (1) = ?
4. Find V (x) such that y (x) = e4x V (x) is a particular solution of the differential equation d2y/dx2 - 8dy/dx + 16 y = 2x + 11x10 + 21x20) e4x at x → 5:.........................
5. Solve the differential equation xy.dy/dx = 3y2 + x2 with the initial condition y = 2, when x = 1, then y (5):...................
6. y1 + xy = f (x), where f is Integrable then find y (x):....................................
7. Find the solution of D. E
dy/dx + y.dΦ/dx= Φ(x) (dΦ)/dx
8. y1 = ez and y = e-z are the solution of homogenous D. E. of the X2 (d2y)/dx2 + n dy/dx y = 4x log x
Then the find the solution of above differential equation:....
9. Let y1 (x) & y2 (x) be the L. I. solution if y" + f(x) y' t Q (x) y = R (x) & W (x) = y1y2'- y2y1' And ∃ a point x1 s. t. W (x1) = 0 then find the value of W (x) in [x1 - ∈, x1 + ∈] where ∈ > 0
10. The equation of the curve satisfying sin y dy/dx = cos y (1- x cos y) and passing through the origin is secy = x + 1..........
11. Solve the initial value problem
Y' - y + y2 (x2 + 2x + 1) = 0, y (0) = 1.......
12. Solve the differential equation
y' + xy = y1/2 e-x2/4 sec x.
13. Solve (x2 + 1)/y2 dy/dx - 5 (x2 -1) = 4x/y...............
14. Find a function Φ (x, y, z) such that dΦ (x, y, z) = yzdx + (z + xz + z2)dy + (y + xy + 2yz) dz and Φ (0, 1, -1) = 0............................
15. Let y (x) be the solution of the initial value problem
X2y" + xy' + y = x, y (1) = y' (1) = 1
Then the value of y ( eΠ/2) is ................................
16. If D ≡d/dx then the value of 1/((xD+1)) (x-1) is at x = 1..............
17. Let y1 (x) and y2 (x) be two solutions of (1 - x2) (d2 y)/dx2 -2x dy/dx + (sec x) y = 0 with Wronskian W (x). If y1 (0) = 1(dy1/dx)x=0 = 0 and W(1/2) = 1/3, then (dy2/dx)x=0 equals:...........
18. If y (x) is the solution of the differential equationdy/dx = 2 (1+y) √y satisfying y (0) = 0; Π/2 = 1:
Then the largest interval ( to the right of origin) on which the solution exists is
19. A particular solution of x2 (d2 y)/dx2 + 2x dy/dx + y/4 = 1/√x is at x = 1..................
20. Consider the differential equation dy/dx =5y - 6y2 and y0 = y0 As x →+∞, the solution y (x) tends to :..................