The differential equation dydxpy q are function of x only

Question

Differential Equation

1. If y₁ (x), y₂ (x)are solution of y" + xy' + (1-x²)y = sin x, then which of the following is also its solution:..................
(a) y₁ (x) + y₂ (x) (b) y₁ (x) - y₂ (x) (c) 2y₁ (x) - y₂ (x) (d) y₁ (x) - 2y₂ (x)

2. The differential equation dy/dx+py = Q are function of x only, have integrating factor
(a) e^fPdx (b) e^fQdy (c) e^fPdy (d) e^fQdx

3. The differential equation whose linearly independent solutions are cos 2x, sin 2x, and e^-xis:.........................
(a) (D³ + D² + 4D) y= 0 (b) (D³ - D² + 4D - 4) y= 0 (c) (D³ + D² -- 4D - 4) y= 0 (d) N.O.T

4. The differential equation dy/dx = k (a - y) (b - y) when solved with the condition y (0) = 0, fields the result..................
(a) (b (a-y))/(b (b-y)) = e^(a-b)kx (b) (a (b-y))/(b (a-y)) = e^{(a- b)kx} (c) (b (a-x))/(a (b-x)) = e^{(b- a)kx} (d) xy = kc

5. The solution of (x+1) dy/dx = 1- e^-y is.................
(a) (x + 1) ey = x + c (b) (x + 1) (1+ey ) = c (c) (x+1)/1+e^x = c (d) (x+1)/1+e^y = c

6. (D+1) x + (D-1)y = e^t (D² + D + 1)x + (D² - D + 1)y = t², where D = d/dt:........
(a) y = 1/3 (3t + t² - 2e^t)  (b) y = 1/4 (2t + t³ - 3e^t) (c) y = 1/2 (2t + t² - 2e^t) (d) y = 1/2 (2t + t² - 3e^t)

7. Integrating factor of DE (y² + 2x²y) dx + (2x³ - xy) dy = 0 is............................................
(a) x^1/2 y^-5/2 (b) x^3/2/y^5/2 (c) x^-5/2 y^-1/2 (d) x^-5/2 y^-5/2

8. If e^ax 4 (x) is a PI of d²y/dx² - 2a dy/dx + a2y = f (x), where a is a constant, then (d²y)/dx² is equal to:...
(a) f (x) (b) f(x) e^ax (c) f(x) e^-ax (d) f(x) (e^ax + e^-ax)

9. Any equal contains n-arbitrary constants, then the order of differential equation derived form it is : .............................................
(a) n (b) n-1 (c) 2 (d) n+1

10. The solution fo differential equation y' sin x = y log y satisfying initials condition y (Π/2) e is:......................
(a) etan x/2 (b) ecot x/2 (c) log tan (x/2) (d) log cot (x/2)

11. Let y = ex be a solution of x (d²y/dx² - dy/dx + (1- x) y = 0, then the second linerarly independent solution of this ODE is:................
(a) xe^-x + 1/2 (b) 1/2 (x- 1/2) e^-x (c) - 1/2 (x+1/2) (d) xe^-x - 1/2

12. Let y be function of x satisfying dy/dx = 2 x³√y - 4xy of y (0) = 0 then y (1) is:...................
(a) 1/4e² (b) 1/e (c) e^1/2 (d) e^3/2

13. The IF of DE (2x y⁴ e^y + 2xy³ + y) dx + (x² y⁴ e^y - x² y² - 3x)dy = 0:..........................................
(a) 1/x⁴ (b) y²/x⁴ (c) 1/y⁴ (d) N.O.T

14. Particular integral of the following differential equation (D² - 4D +4) y = (e^2x log x)/x PS obtain by integrating
(a) (x log x)/2  (b) (x(log x)²/2  (c) (log x)²/2 (d) ((x logx)²)/2

15. The solution of the differential equation xdy - ydx = √(x² + y² ) dx is :...............................

(a) y + √(x² + y² ) = xc (b) y + √(x²+ y² ) = x²c (c) y -√(x²+ y² ) = xc² (d) N.O.T

16. If y₁ and y₂ are two solutions of y" + p (x) y' + q (x) y = 0, then general solution of this equation y₁and y₂ are :........
(a) Linearly dependent (b) linearly independent (c) proportional

17. The solution of the differential equation (x³ y³ + xy) dy/dx = ? is:.............................................
(a) 1/x = 2^-y2/2 + Ae^y2/2  (b) x = 2 - y² + Ae^y2/2 (c) 1/x = 2-y² + Ae^-y2/2 (d) N.O.T

18. Solution of (d²y/dx² - 2/x dy/dx + (1+2/x² )y = xe^x is
(a) y = x (c₁ cos x + c₂ log x + 1/2ex)
(b) y = x (c₁ cos x + c₂ sin x )
(c) y = - x (c₁ cos x + c₂ sin x )
(d) y = x (c₁ cos x + c₂ sin x + 1/2ex)

19. Solution of DE (d²y)/dx² + y = cosec x is
(a) y = c₁ cos x + c₂ sin x - x cos x + sin x log sin x
(b) y = c₁ sin x + c₂ cos x - x sin x + sin x log cos x
(c) y = c₁ cos x + c₂ sin x-x sin x + cos x log sinx (d) N.O.T

20. A function f (t) = d/dt {dx/(1-cos t cos x)}satisfies the differential equation :................................
(a) dt/dt + 2f (t) cos t = 0 (b) dt/dt - 2f (t) cos t = 0 (c) dt/dt + 2f (t) sin t = 0 (d) dt/dt - 2f (t)sin t = 0

21. Solution of (d²y)/dx² + cot x dy/dx + 4y cosec²x= 0 is:........
(a) y = k₁ cos (2 log tan x/2 + k₂ )
(b) y = k₁ sin ( log tan x/2 + k₂ )
(c) y = k₁ cos ( log tan x/2- k₂ )
(d) y = k₁ cos (-log tan x/2 + k₂ )

22. Given, an equation (d³y)/dx³ + d²y/dx² + 4 dy/dx + 4 = 0. Has solution:....................................................
(a) y = c₁e^-x + xc₂ e^-x + x² c₃e^-x
(b) y = c₁ cos 2x + c₂ sin 2x
(c) y = c₁e^-x + c₂ sin 2x + c₃ sin 2x (d) N.O.T

23. The integrating factor for the differential equation (x + 1) dy/dx - y = e^3x (x + 1)² is:............
(a) 1/(x+1) (b) x + 1 (c) 1/(x²+1) (d) x²+1

24. The solution of the differential equation x {y (d²y/dx² +(dy/dx)² } + y dy/dx = 0 , is :........................
(a) ax + by = c (b) ax² + by = 0 (c) ax² + by² = 0 (d) ax + by² = 0

25. If y' - x ≠ 0, a solution of differential equation y' (y'+y) = x (x + y) is given if y(0) = 0
(a) 1 - x - e^-x (b) 1- x +e^x (c) 1+ x +e^-x (d) 1+ x +e^x

26. The solution of the differential equation (cos x - sin x) (d²y)/dx² + 2sin x dy/dx - (cos x + sin x) y = 0, given that y = sin x is a solution, is :..........
(a) c₁ cos x + c₂ x sin x (b) c₁ sin x + c₂ x sin x (c) c₁ sin x + c₂ ex (d) ) c₁ sin x + c₂x

27. LetΦ₁(x) and Φ₂(x) are particular integral of L(y) = e^axf(x)a, b being constants, then a PI of L(y) = 2 be^ax is:..............
(a) bΦ₁(x) + Φ₂(x)  (b) Φ₁(x) - bΦ₂(x) (c) aΦ₁(x) + bΦ₂(x) (d) b[Φ₁(x) + Φ₂(x)]

28. The particular solution of (d³y)/dx³ + y = cos (2x - 1) is:.........................
(a) 1/65 [cos (2x - 1) - 8] (b) 1/65 [cos (2x - 1) - 8 sin (2x - 1)] (c) 1/65 [cos (2x + 1) + 8] (d) N.O.T

29. Let W [y₁ (x), y₂ (x)] is the wonkier formed for the solution y₁ (x) and y₂ (x) of the differential equation y" + a₁ y'+ a₂y = 0, if W ≠ 0 for same x = x0 in [a, b], then:..................
(a) It vanishes for any x ∈ [a, b]
(b) it does not vanish for any x ∈ [a, b]
(c) If does not vanish only at x = a
(d) If does not vanish only at x = b
30. If k is any constant such that xy + k = e ((x-1)²)/2 satisfies the differential equation x dy/dx = (x² - x - 1) y + (x - 1) then k is equal to :.......
(a) 1 (b) 0 (c) -1 (d) -2

31. All non- trivial solution of x2 y" + xy' + 4xy = 0, x > 0 are:....................................................
(a) bounded and non- periodic
(b) unbounded and non periodic
(c) bounded periodic
(d) unbounded and periodic

32. Ly= x^ex log x [x > 0], when L = (d²/dx² + P.d/dx + Q) the two LI solution of Ly = 0, are xe^x and ex, then PI is:........
(a) xe^x[x/2 log (x-x)²/x] + e^-x [-x³/3 log x³/a]
(b) x²e^x [x²/2 log (x-x)²/x] + ex [-x²/2 log (x + x³)/a]
(c) x²ex[x²/2 log (x-x)²/x] + ex[-x³/3 log (x + x³)/a].............
(d) N.O.T

33. The solution of y" + ay' + by = 0. Where and b are constants, approaches to zero as x →∞ then:.........
(a) a > 0, b > 0 (b) a > 0, b < 0 (c) a < 0, b < 0 (d) a < 0, b > 0

34. Differential equation |y'|+|y| = has.....................................
(a) G S solution
(b) G S solution but no arbitrary constant
(c) Particular solution which is bounded
(d) N.O.T

35. The ordinary differential equation x dy/dx - y = = 2x² with initial condition y (0) = 0, has:.....
(a) no solution (b) a unique solution (c) two district solution (d) Infinitely many solutions

36. Let y₁ (x) and y₂ (x) be two linearly independent solution at the differential equation x (d²y)/dx² - 2x² dy/dx + e^xy = 0 satisfying y₁ (0) = 1 y₂ (0) = - 1 y₁' (0) = 1 and y₂' (0) = 1, then the wronskian of y₁ (x) and y₂ (x) at x = 2............ w|y₁ y₂ | x = ?equal to:...............
(a) 2e^-1 (b) 2e² (c) 2e⁴ (d) 2e^-4

37. Let y (x) be 0 non-trivial solution of the 2nd order linear differential equation (d² y)/dx² + 2c dy/dx + ky = 0, where c < 0, k > 0 and c2 > k, then
(a) |y(x)|→∞ as →∞(b) |y(x)|→0 as →∞ (c) (x→lim )±∞ |y(x)| exists and is finit(d) N.O.T

38. If y₁ (x) and y₂ (x) are solution of y" + x²y' - (1- x) y = 0 such that y₁ (0) = 0, y₁' (0) = - 1 and y₂(0) = - 1, y₂' (0) = 1, the wronskian w (y₁, y₂) on R:.......................................................
(a) is never zero (b) is identically
(c) is zero only finite member of points
(d) is zero at countable infinite member of points

39. Let y₁ and y₂ be two linearly independent solutions of y" + (sin x) y = 0, 0 ≤ x ≤ 1 Let g (x) = w (y₁, y₂) (x) be the Wronskian of y₁ and y₂. Then:................................................................
(a) g' > 0 on (0, 1) (b) g' < 0 on (0, 1)
(c) g' vanishes at any one point
(d) g' vanishes at all points of (0, 1)

40. The Largest value of c such that there exists a function h (x) for - c < x < c that is a solution dy/dx = 1 + y² with h (0) = 0 is given by
(a)Π/3 (b) Π/2 (c) Π/4 (d)Π

41. If 2x (1 - y) = k and g (x, y) = L are orthogonal families of curves where K and L constant then g (x, y) is
(a) x² + 2y - y² (b) 2 {y +2 [y + (1- x)] (c) x² + 2x - y² (d) x² - 2y +y²

42. If y₁ (x) = x and y₂ (x) = xe^x are two linearly independent solutions of x² (d²y)/dx² - x (x + 2)dy/dx + ( x+ 2)y = 0, then the interval on which they form a fundamental set of solution is:.............
(a) x > 0 or x < 0 (b) -1 < x < ∞ (c) -1 < x < z (d) - ∞ < x < ∞

43. The solution of the differential equation y dy/dx + (1 + y²) sin x = 0 y (0) = 0, exists in the open interval (-a,a), then a is equal to :.......
(a) 2 cos^-1 (1/2 √log2) (b) 2 tan^-1(1/2 √log2) (c) 2 sin^-1 (1/2 √log2) (d) 2 cot^-1 (1/2 √log2)

44. For non-homogeneous equation y'+ P(x) y = r(x), if y₁ an d y₂ are its solution, then the solution is homogenous equation y' + P (x) y = 0 is:..................
(a) y = y₁ - y₂ (b) y = y₁/y₂ (c) y = y₂/y₁ (d) N.O.T

45. Solve dy/dt = √(|y| ), for 0 (a) has unique solution (b) has no solution
(c) has two independent solution
(d) has infinite solution

46. The general solution dy/dx + y = f (x), where

(d) N.O.T

47. For (d²y)/dx² + 4y = tan 2x solving by vacation of parameter the value of wronskian W is:...
(a) 1 (b) 2 (c) 3 (d) 4

48. Let a, b∈ R, let y = (y₁, y₂ ) T be a solution of the system equation y₁¹ = y₂¹, y₂ = ay₁ + by₂ every solution (x) → 0 as n → ∞ if :............
(a) a < 0, b < 0 (b) a > 0, b>0 (c) a> 0, b > 0 (d) a > 0, b >0

49. Suppose y_p (x) = x cos 2 (x) is a particular solution y" + αy = - 4 sin 2x, then the constant α equals:..............................................
(a) - 4 (b) - 2 (c) 2 (d) 4

50. The value of wronksion W (x, x², x³) is:......
(a)2x⁴ (b) 2x² (c) 2x³ (d) N.O.T

51. Which of the following transformation reduce the differential equation dz/dx + z/x log z = z/x (log z)² into the form du/dx + P (x) ..
(a) u = log z (b) u = 1/(log z) (c) u = e^z (d) u = (logz)z

52. If (c₁ + c₂ log x)/x is the general solution of the differential equation x² d²y/dx² + kx dy/dx + y = 0, x > 0 then k equals:......
(a) 3 (b) - 3 (c) 2 (d) - 1

53. Consider the differential equation dy/dx = ay - by² where a, b > 0, and y (0) = y₀ as x → ∞, the solution y (x) finds to:.......
(a) 0 (b) a/b (c) b/a (d) y0

54. If g (x, y) dx + (x + y) dy = 0 is an exact differential equation and if g (x, 0) = x², then the general solution of the differential equation is :....
(a) 2x³ + 2xy + y² = c (b) 2x³ + 6xy + 3y² = c (c) 2x + 2xy + y² = c (d) x² + 2xy + y² = c

55. Let f₁ g: [- 1, 1] → R, f (x) = x³, g (x) = x² |x|. Then:......
(a) f and y are linearly independent on [ -1, 1]
(b) f and y are linearly dependent on [ -1, 1]
(c) f (x) g' (x) - f' (x) g (x) is not identical zero on [ - 1, 1]
(d) Then exists continuous function p (x) and g (x) such that

56. Let k be a real constant. The solution of the differential equation dy/dx = 2y + z and dz/dx = 3y satisfies the relation :..........
(a) y - z = ke^3z (b) y + z = ke^-x (c) 3y + z = ke^3z (d) 3y + z = ke^-3x

57. Let y' (x) + y (x) g' (x) = g (x) g'(x) if y (0) = 0 & g(0) = 0, then f (2) = ?
(a) 0 (b) 3 (c) 5 (d) N.O.T

58. Let f : [1, ∞] →[2, ∞] be a differentiable function such that f (1) = 2, 6 ₁∫^xf (t). dt = 3x f (x) - x³ for all x ≥ 1, then the value of f(z):...
(a) 8 (b) 6 (c) 3 (d) N.O.T

59. If x = 1 + dy/dx + 1/L² (dy/dx)² +1/L³ (dy/dx)³ +.......+ 1/Lⁿ (dy/dx)ⁿ+......., degree of differential equation
(a) not define (b) n (c) 1 (d) N.O.T

60. The O. T. of the circle x² + y² - ay = 0 if y (1), where a is parameter.
(a) x² + y² = cx (b) x² + y² = 2cx (c) x² + y² = c (d) x² + y² = cx

61. The particular solution of the differential equation y" + y' + 3y = 5 cos (2x + 3) is:............
(a) 2 cos (2x + 3) - sin (2x + 3) and two absolute constant. (b) 2 sin (2x + 3) - cos (2x + 3) (c) (a) & (b) (d) N.O.T

62. If n is a positive integral value of the 1/((D-α)) . e^αx:.........
(a)xⁿ/L⁽ⁿ⁺¹⁾ e^αx (b) 1/Lⁿ e^αx (c) (a) & (b) (d) N.O.T

63. According to E. C. E the P. I if (x² D² + 3xD +1) y = 1/(1-x)² :......................
(a) (c₁ + c₂ log x) x^-1 (b) x^-1 log {x|(1-x)} (c) (a) & (b) (d) N.O.T

64. Solve (x² + y² + x)dx - (2x² + 2y² - y) dy = 0
(a) x - 2y + 3/2 log (x² + y²) = c  (b) x - 2y + 1/2 log (x² + y²) = c (c) (a) & (b) (d) N.O.T

65. If (D⁴ + m⁴) y = 0 then the solution of the equation:..........
(a) m^{ei Π/4} (b) m (c) m^{e5i Π/4} (d) m e^{7i Π/4}

66. The solution of D. E (d²y)/dx²- y = 1 which vanishes when x = 0 and tends to a finite limit as x→∞:..
(a) y = e2x^-1 (b) y = ex^-1 (c) y = ex (d) N.O.T

67. The set of eigen vectors x₁,x₂,......., x_n corresponding to distinct eigen vectors λ₁,λ₂,.....λ_n then square of matrix
(a) singular (b) L.D (c) (a) & (b) (d) N.O.T

68. If A is any non-singular square matrix and |A| = 5 and order of A is 4 then determinant of adj (adjA) = ?
(a) 1.9 × 10⁶ (b) 5.9 (c) 3220 (d) N.O.T

69. If K is any scalar then adj (K In):..................
(a) K^n-2 In (b) K adj In (c) (a) & (b) (d) N.O.T

70. Let Abe a 3x3 matrix with trace (A) = 3, and det (A) = 2. If 1 is Eigen vector, then other two eigen values:...................................
(a) 1+ I (b) 1- I (c) (a) & (b) (d) N.O.T