MTH401 — Differential Equations — Midterm Past Papers & Solved MCQs

Q1. Which of the following is the first order linear equation in unknown variable y?

• A.z dy/dx + (sinx)y = cosx✓ Correct
• B.y dx/dy + (siny)x = cosy
• C.y dx/dy + (siny)x = cosx
• D.y dx/dy + (sinx)x = cosy

Q2. Which of the following function would satisfy: dy/dx = d^2y/dx^2 = ... = d^ny/dx^n?

• A.y = Ae^x✓ Correct
• B.y = Axe^x
• C.y = Ae^{xx}
• D.y = Ax^x e^{xx}

Q3. If y=2+x, then which of the following is true for it?

• A.Its annihilator is D
• B.Its annihilator operator is D^2✓ Correct
• C.Its annihilator operator is D^3
• D.Its annihilator operator is D+1

Q4. For the non-exact differential equation M(x,y)dx+N(x,y)dy=0, if (∂M/∂y - ∂N/∂x)/N is a function of x, then the integrating factor is:

• A.function of x also✓ Correct
• B.function of y
• C.constant
• D.multi-variable function of both x and y

Q5. Which of the following are explicit solution of the differential equation: dy/dx = -x/y?

• A.y = ±√(4+x)
• B.y = ±√(-4+x^2)
• C.y = ±√(4-x^2)✓ Correct
• D.y = ±√(-4-x^2)

Q6. For f(x,y)=2x/(3y)+7, f(tx,ty)=---

• A.f(x,y)
• B.t f(x,y)✓ Correct
• C.t^2 f(x,y)
• D.t^3 f(x,y)

Q7. What is the annihilator operator of the function g(x)=4sinx?

• A.(D^2-1)
• B.(D^2+4)
• C.(D^2-4)
• D.(D^2+1)✓ Correct

Q8. The differential equation dy/dx - y = y^3 is:

• A.Bernoulli's✓ Correct
• B.Homogeneous
• C.Cauchy
• D.Bessel

Q9. The integrating factor for the first order linear differential equation: dy/dx + ycotx = sin^2x is ______.

• A.sinx✓ Correct
• B.cosx
• C.e^{sinx}
• D.e^{cosx}

Q10. In exponential model for the population growth: P(t)=P0 e^{kt}. If k>0, then lim t→∞ P(t) =

• A.0
• B.1
• C.∞✓ Correct
• D.-∞

Q11. The hook's law states that the force F is proportional to the _________.

• A.Length
• B.Elongation✓ Correct
• C.Weight
• D.None of these

Q12. Separable form of the differential equation: dy/dx = y-1 is ________, where v=y-1.

• A.dx/x = dv
• B.dx = vdv
• C.dy/v = dx✓ Correct
• D.dv/v = dx/x

Q13. Which of the following is an equivalent form of the exact differential equation: ydx+xdy=0?

• A.d(x/y)=0
• B.d(y/x)=0
• C.d(xy)=0✓ Correct
• D.d(x+y)=0

Q14. In the general solution of a separable differential equation is sin^{-1}y = cos^{-1}x + c, provided that y(1/√2)=1/√2, then c = ---.

• A.π/2
• B.π/4
• C.-π/2
• D.0✓ Correct

Q15. The differential equation of orthogonal trajectory to the family of curves x-2y=c is:

• A.dy/dx = -1/2
• B.dy/dx = 1/2
• C.dy/dx = 2
• D.dy/dx = -2✓ Correct

Q16. Classify the following differential equation e^x dy/dx + 2y = 3xy

• A.Separable and not linear
• B.Linear and not separable
• C.Both separable and linear✓ Correct
• D.Neither separable nor linear

Q17. The periodic time is given by:

• A.w/(2π)
• B.2π/w✓ Correct
• C.2π+w
• D.π/w

Q18. If x^2 y^3 dx + x^3 y^2 dy = 0 has the equivalent form as d(1/3 x^3 y^3)=0, then its solution is _____.

• A.x^3+y^3=c
• B.x^3-y^3=c
• C.x^3 y^3=c✓ Correct
• D.x^3/y^3=c

Q19. The family of parabolas y^2=4ax are solutions of the differential equation: dy/dx = 2a/y for ------- value(s) of a.

• A.infinite✓ Correct
• B.finite
• C.unique
• D.no

Q20. If y=c1 e^{(-2+√6)x} + c2 e^{(-2-√6)x} is the complementary solution of:

• A.Ax+B
• B.Ax^2+Bx+C
• C.Ax^3+Bx^2+Cx+D✓ Correct
• D.(Ax^2+Bx)2x^2

Q21. If initial amount of radioactive isotope is 100g. What will be the amount at the end of 30 days such that K=0.043?

• A.371.415
• B.380.560
• C.363.297✓ Correct
• D.360.351

Q22. If 2,3,5 are real roots of a differential equation, then the general solution is ________.

• A.yc = c1 e^{2x} + c2 e^{3x} + c3 e^{5x}✓ Correct
• B.yc = c1 e^{2x} + c2 e^{-3x} + c3 e^{5x}
• C.yc = c1 e^{2x} + c1 e^{-3x} + c1 e^{5x}
• D.yc = (c1+c2+c3)e^{3x}+e^{2x}

Q23. Wronskian W(x,2x) = __________.

• A.0✓ Correct
• B.1
• C.-1
• D.None of these

Q24. Which of the following is the implicit solution of the differential equation: dy/dx = -x/y?

• A.x+y+4=0
• B.x^2+y^2-4=0✓ Correct
• C.x^2-y^2+4=0
• D.x^2-y^2-4=0

Q25. ydx - y(sinx)dy = 0, is an example of ______ differential equation.

• A.Exact✓ Correct
• B.Non-exact
• C.Non-linear
• D.Non-homogeneous

Q26. Which of the following is an example of ordinary differential equation?

• A.(y-x)dx+4xdy=0✓ Correct
• B.∂u/∂y = -∂u/∂x
• C.ax^2+bx+c=0, a≠0
• D.φ(x)=f(x)+λ∫k(x,t)φ(t)dt

Q27. A differential equation M(x,y)dx+N(x,y)dy=0 is exact if there exists a multi-variable function f(x,y) such that _______.

• A.df(x,y)=∂f/∂x dx+∂f/∂y dy✓ Correct
• B.∫f(x,y)dx=∫∂f/∂x dx+∫∂f/∂y dy
• C.f(x,y)=∂f/∂x dx+∂f/∂y dy
• D.f(x,y)=∫∂f/∂x dx+∫∂f/∂y dy

Q28. Which of the following is an initial value problem (IVP)?

• A.d^2y/dx^2+y=0
• B.d^2y/dx^2+y=0, x∈[-2,2]
• C.d^2y/dx^2+y=0, y(1)=-2 and y'(1)=-2✓ Correct
• D.d^2y/dx^2+y=0, y(1)=-2 and y'(π/2)=-2

Q29. Which of the following would be a constant solution of the separable differential equation: dy/dx = e^{x+y}?

• A.y=0✓ Correct
• B.y=1
• C.y=a∈R
• D.No constant solution exist

Q30. The family of curve x^2+y^2=C^2 represents a family of circles centered at ___________.

• A.(1,0)
• B.(0,1)
• C.(0,0)✓ Correct
• D.(1,1)

Q31. The differential equation: (x+2y^3) dy/dx = y is linear in unknown:

• A.variable y
• B.multi-variable x and y
• C.variable x✓ Correct
• D.dy/dx

Q32. For a system in simple harmonic motion which of the following is the time required to complete a cycle of motion?

• A.Frequency
• B.Amplitude
• C.Period✓ Correct
• D.Revolution

Q33. Suppose that y1, y2, ..., yn are n solution of the homogeneous nth order differential equation. Then the solution is linearly dependent on an interval if and only if _________.

• A.W(y1,y2,...,yn)=0✓ Correct
• B.W(y1,y2,...,yn)=∞
• C.W(y1,y2,...,yn)≠0
• D.W(y1,y2,...,yn)=-∞

Q34. If we substitute u=y/x in differential equation.

• A.u e^u du = dx
• B.u e^{-u} du = dx✓ Correct
• C.u e^u du = dy
• D.u e^{-u} du = dy

Q35. Which of the following would be a particular solution of the differential equation: dy/dx = 4?

• A.y=4x+a
• B.y=ax+4
• C.y=ax+a
• D.y=4x+4✓ Correct

Q36. Separable form f(y)dy+g(x)dx=0, of the differential equation x siny dx + (x^2+1) cosy dy = 0 is ________.

• A.tany dy + x/(x^2+1) dx = 0
• B.coty dy + x/(x^2+1) dx = 0✓ Correct
• C.tany dy + x/(x^2-1) dx = 0
• D.coty dy + x/(x^2-1) dx = 0

Q37. The differential equation (1+lnxy)dx + (1+x/y)dy = 0 is exact because __________.

• A.∂M/∂y = ∂N/∂x = 1/x
• B.∂M/∂x = ∂N/∂y = 1/y
• C.∂M/∂y = ∂N/∂x = 1/y✓ Correct
• D.∂M/∂x = ∂N/∂y = 1/x

Q38. The differential equation (3x^2y+3)dx+(x^2y+2)dy=0 is

• A.Exact✓ Correct
• B.Linear
• C.Homogenous
• D.Separable

Q39. The assumed particular solution for the U.C differential equation y''-2y'=x^2e^{2x} is

• A.y_p=c_1e^{2x}+c_2x
• B.y_p=(Ax+B)e^{2x}
• C.y_p=(Ax^2+Bx+C)e^{2x}✓ Correct
• D.None of these

Q40. The differential equation dy/dx + (2/x)y = ln x is an example of

• A.Separable
• B.Homogenous
• C.Exact
• D.None of these✓ Correct

Q41. For the differential equation y' - 2xy = x, integrating factor is

• A.x^{-2}
• B.e^{x^2}
• C.e^{-x^2}✓ Correct
• D.x^2

Q42. (dy/dx) = (x+3y-5)/(x-y-1). Identify the ordinary differential equation

• A.Homogenous
• B.Separable
• C.Exact
• D.None of these✓ Correct

Q43. In the homogeneous differential equation after substitution v=y/x the equation reduces to.

• A.Separable differential equation.✓ Correct
• B.Exact differential equation.
• C.Remain homogeneous equation.
• D.None of the other

Q44. If the Wronskian W of three function f(x),g(x),h(x) is zero, what can be said about the dependency of the functions

• A.May or may not be dependent✓ Correct
• B.Always dependent
• C.Never dependent
• D.None of the other

Q45. If a_n(x)=0 for some x in I then

• A.I is correct only
• B.I and II are correct✓ Correct
• C.I and III are correct
• D.IV is correct only

Q46. Equation of the form dy/dx + y = x^2 y^2 is called

• A.First order linear differential equation
• B.Bernoulli equation✓ Correct
• C.Separable equation
• D.None of the other.

Q47. If the variation of the path of the curves can be described by the concept of differential equations y axis then which of the following differential equation describe the path for .

• A.dy/dx = 1
• B.dy/dx = 0✓ Correct
• C.dy/dx = -1
• D.dy/dx = ∞

Q48. Suggestive form of the constant input function for the non homogeneous differential equation under the method entitled as 'Method of the undetermined coefficient' is

• A.f(x)=e^x
• B.f(x)=a✓ Correct
• C.f(x)=e^{ax}(Acosx+Bsinx)
• D.Suggestive form is impossible.

Q49. Which of the following function is linearly dependant to the exponential function e^x ?

• A.-e^x✓ Correct
• B.e^{-x}
• C.xe^x
• D.-xe^{-x}

Q50. Eigen values for the system of the differential equations X' = AX are evaluated for the

• A.Solution vector X
• B.Coefficient matrix A✓ Correct
• C.Differentiated solution vector X'
• D.Transpose of the Coefficient matrix A

Q51. The differential equation (3x^2 y + 2)dx + (x^3 + y)dy = 0 is

• A.Exact✓ Correct
• B.Linear
• C.Homogenous
• D.Separable

Q52. The assumed particular solution for the U.C differential equation y' - y = x^2 e^{2x} is

• A.y_p = c1 e^{x^2} + c2 x^2
• B.y_p = (Ax+B)e^{2x}
• C.y_p = (Ax^2+Bx+c)e^{2x}✓ Correct
• D.None of these.

Q53. The differential equation x dy/dx + y = y^2 ln x is an example of

• A.Separable
• B.Homogenous
• C.Exact
• D.None of these.✓ Correct

Q54. The differential equation (x+y)(x-y)dx + x(x-2y)dy = 0 is

• A.Exact✓ Correct
• B.Not exact
• C.Linear
• D.Homogeneous

Q55. The integrating factor of the differential equation (2y^2+3x)dx + 2xydy = 0 is

• A.x✓ Correct
• B.y
• C.xy
• D.1/x

Q56. The form of the particular solution for the differential equation y'' - y = cos x + e^x is:

• A.y_p = Ae^x + B cos x + C sin x✓ Correct
• B.y_p = A e^x + B cos x
• C.y_p = A e^x + B sin x
• D.y_p = A x e^x + B cos x + C sin x

Q57. Determine which of the given functions are linearly independent. f1(x)=x, f2(x)=x^2, f3(x)=4x-3x^2

• A.Linearly independent
• B.Linearly dependent✓ Correct
• C.Cannot be determined
• D.None

Q58. The differential operator that annihilates 4e^(2x) is:

• A.(D-2)(D+5)✓ Correct
• B.(D-2)
• C.(D+2)
• D.(D-2)(D-5)