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) y^n-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 = e^x 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 e^2x + 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/dθ_θ∫^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 e^x+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 ex^2_/2 (b) ke^y2/2 ) (c) k e^xy/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 a_n xⁿ + a_n-1 x^n-1+.....+ a₁x + a₀ have negative real parts. If u (t) is any solution to the ordinary differential equation. a_n (dⁿu)/dtⁿ + a_n-1 (d^n-1u)/dt^n-1+.....+a₁ (du )/dt + a₀u= 0, then lim_t→∞ 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 = e^2x and y₂ = xe^2x, 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) = e^x + e^-x is a D. E. then:............
(a) P.I = e^x/2 + e^x/14 (b) P.I = e^x/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 x₀simultaneously.
(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/ b^2λ = 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 + e^x at x →-∞:...............................................................................
(a) 1/3 (1 + e^-x) (b) 1/3 + 1/6 e^x (c) 1/3 - 1/6 e^x (d) N.O.T

8. x² D² y + 4x Dy + 2y = e^x 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 e^x sin x:......................
(a) W = e^2x (b) W = e^x
(c) A = - e^x (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) = e^4x V (x) is a particular solution of the differential equation d²y/dx² - 8dy/dx + 16 y = 2x + 11x¹⁰ + 21x²⁰) e^4x 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 = e^z 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 = y^1/2 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 ( e^Π/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 :...............