Ordinary Differential Equations MCQ

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• While solving an Ordinary Differential Equation using the unilateral Laplace Transform, it is possible to solve if there is no function in the right hand side of the equation in standard form and if the initial conditions are zero.
  ক

  True

  খ

  False
• For the Transient analysis of a circuit with capacitors, inductors, resistors, we use bilateral Laplace Transformation to solve the equation obtained from the Kirchoff’s current/voltage law.
  ক

  True

  খ

  False
• With the help of _____________________ Mr.Melin gave inverse laplace transformation formula.
  ক

  Theory of calculus

  খ

  Theory of probability

  গ

  Theory of statistics

  ঘ

  Theory of residues
• While solving the ordinary differential equation using unilateral laplace transform, we consider the initial conditions of the system.
  ক

  True

  খ

  False
• Gamma function is said to be as Euler’s integral of second kind.
  ক

  True

  খ

  False
• Which of the following is not a definition of Gamma function?
  ক

  \(\Gamma(n) = n!\)

  খ

  \(\Gamma(n) = \int_{0}^{\infty} x^{n-1} e^{-x}dx\)

  গ

  \(\Gamma(n+1) = n\Gamma(n)\)

  ঘ

  \(\Gamma(n) = \int_{0}^{1} log \left({1 \atop y}\right)^{n-1}\)
• Using the method of undetermined coefficients find the P.I for the D.E x’’’(t) – x’’(t) = 3et + sin⁡t.
  ক

  xp = 3et + \(\frac{t}{2}\) (cos⁡t – 2 sin⁡t )

  খ

  xp = 3tet + \(\frac{1}{2}\) (cos⁡t + sin⁡t )

  গ

  xp = 3tet + \(\frac{t}{3}\) (4cos⁡t + 2sin⁡t )

  ঘ

  xp = 3et + \(\frac{1}{2}\) (cos⁡t – sin⁡t )
• Find the Particular integral solution of the D.E (D2 – 4D + 3)y = 20 cos x by the method of undetermined coefficients.
  ক

  yp = 4 cos⁡x – 3 sin⁡x

  খ

  yp = 2 sin⁡x – 4 cos⁡x

  গ

  yp = – 3 cos⁡x + 4 sin⁡x

  ঘ

  yp = 2 cos⁡x – 4 sin⁡x
• Solution of the D.E y’’ + 3y’ + 2y = 12x2 when solved using the method of undetermined coefficients is ________
  ক

  y = c1 ex + c2 e2x + 2 – 11x + x2

  খ

  y = c1 e – x + c2 e – 2x + 18 + 21x + 3x2

  গ

  y = c1 ex + c2 e – 2x + 11 + 18x + 2x2

  ঘ

  y = c1 e – x + c2 e – 2x + 21 – 18x + 6x2
• Solution of the D.E y’’ – 4y’ + 4y = ex when solved using method of undetermined coefficients is _____
  ক

  y = (c1 + c2)e2x + 2ex – 1

  খ

  y = (c1 + c2 x)e2x + 4ex – 4

  গ

  y = (c1 + c2 x)e2x + ex

  ঘ

  y = (c1 + c2 x)ex + 4ex
• A body in air at 25℃ cools from 100℃ to 75℃ in 1 minute. What is the temperature of the body at the end of 3 minutes? (Take log(1.5)=0.4)
  ক

  40℃

  খ

  47.5℃

  গ

  42.5℃

  ঘ

  50℃
• What is the degree of the differential equation, x3-6x3y3+2xy=0?
  ক

  3

  খ

  5

  গ

  6

  ঘ

  8
• Who invented the rectangular coordinate system?
  ক

  Frans van Schooten

  খ

  René Descartes

  গ

  Isaac Newton

  ঘ

  Gottfried Wilhelm Leibniz
• In the equation, y = x2+c,c is known as the parameter and x and y are known as the main variables.
  ক

  True

  খ

  False
• A rectangular frame is to be made of 240 cm long. Determine the value of the length of the rectangle required to maximize the area.
  ক

  24 cm

  খ

  60 cm

  গ

  240 cm

  ঘ

  120 cm
• What is the differential equation of a family of parabolas with the foci at the origin and axis along the X-axis?
  ক

  2xy’+ 4y(y’)2-y=0

  খ

  xy’+ y(y’)2-y=0

  গ

  2xy’+ y(y’)2-y=0

  ঘ

  2xy’+2y(y’)2-y=0
• In the formation of differential equation by elimination of arbitrary constants, after differentiating the equation with respect to independent variable, the arbitrary constant gets eliminated.
  ক

  False

  খ

  True
• Which of the following is true with respect to formation of differential equation by elimination of arbitrary constants?
  ক

  The given equation should be differentiated with respect to independent variable

  খ

  Elimination of the arbitrary constant by replacing it using derivative

  গ

  If ‘n’ arbitrary constant is present, the given equation should be differentiated ‘n’ number of times

  ঘ

  To eliminate the arbitrary constants, the given equation must be integrated with respect to the dependent variable
• Beta function is not a symmetric function.
  ক

  True

  খ

  False
• Which of the following is one of the criterions for linearity of an equation?
  ক

  The dependent variable and its derivatives should be of second order

  খ

  The dependent variable and its derivatives should not be of same order

  গ

  Each coefficient does not depend on the independent variable

  ঘ

  Each coefficient depends only on the independent variable
• Solution of a differential equation is any function which satisfies the equation.
  ক

  False

  খ

  True
• Which of the following is not a standard method for finding the solutions for differential equations?
  ক

  Variable Separable

  খ

  Homogenous Equation

  গ

  Bernoulli’s Equation

  ঘ

  Orthogonal Method
• Which of the following is not an example of linear differential equation?
  ক

  y=mx+c

  খ

  x+x’=0

  গ

  x+x2=0

  ঘ

  x”+2x=0
• Which one of the following is not a criterion for linearity of an ordinary differential equation?
  ক

  The dependent variable y and its derivatives are of first degree

  খ

  The derivatives of the dependent variable y should be of second degree

  গ

  No product terms of y and/or any of its derivatives are present

  ঘ

  No transcendental functions of y and/or its derivatives occur
• What is the degree of the differential equation, 4x3-6x2 y3+2y=0?
  ক

  3

  খ

  5

  গ

  1

  ঘ

  8
• What is the order of the differential equation, y”+y’-x3y=sinx?
  ক

  2

  খ

  1

  গ

  0

  ঘ

  3
• Which of the following correctly defines ordinary differential equations?
  ক

  A differential equation in which a dependent variable (say ‘y’) depends on only one independent variable (say ’x’)

  খ

  A differential equation in which an independent variable (say ‘y’) depends on only one dependent variable (say ’x’)

  গ

  A differential equation in which a dependent variable (say ‘y’) depends on one or more independent variables (say ’x’, ’t’ etc.)

  ঘ

  A differential equation in which an independent variable (say ‘y’) depends on one or more dependent variables (say ’x’, ’t’ etc.)