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) e^x (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) cx^3/2 (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) e^x 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) = e^x 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) e^3x + 3e^-4y = 4 (b) 4e^3x - e^-4y = 3 (c) 3e^3x + 4e^4y = 7 (d) 4e^3x +3e^-4y = 7

18. P.I of (D² - 2D + 1) y = xe^x sin x is :.........
(a) e^x (x cos x + 2 sin x)
(b) -e^x (2 cos x + x sin x)
(c) -e^x (x cos x + 2 sin x)
(d) N.O.T

19. The orthogonal trajectory of the family of curve a^n-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) e^Φ(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) e^x (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) ((4e^Π/2-6))/5  (b) ((6e^Π/2-4))/5 (c) ((8e^Π/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 + βe^2x)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) xe^3x (b) x²e^3x (c) xe^2x (d) e^3x

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) x^3/2
(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 = e^x y², y (0) = 1Y" - 2y' + αy = 0 Then the value of ∝.............................................................

8. The I. F. (x⁷y² + 3y )dx + (3x⁸ y - x) dy = 0 is x^m 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' + xe^xy = 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 e^t then t lim_→ ∞ e^-t y (t):........................

19. The D. E. y" + 6y' + 9y = 50e^2x 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).....................................................