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Unit 2: Linear differential equation-I - Practice Quiz

MTH174 — Engineering Mathematics 60 Questions
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1 Which of the following is a first-order linear differential equation in ?

Introduction to linear differential equation Easy
A.
B.
C.
D.

2 What is the standard form of a first-order linear differential equation?

Introduction to linear differential equation Easy
A.
B.
C.
D.

3 In a linear differential equation, the dependent variable and its derivatives occur with what power?

Introduction to linear differential equation Easy
A. Second power only
B. Any integer power
C. Zero power only
D. First power only

4 What is the integrating factor for ?

Solution of linear differential equation Easy
A.
B.
C.
D.

5 For , which is the general solution?

Solution of linear differential equation Easy
A.
B.
C.
D.

6 For , the solution is obtained using which factor?

Solution of linear differential equation Easy
A. Separating factor
B. Complementary factor
C. Characteristic factor
D. Integrating factor

7 Two functions and are linearly dependent if one can be written as what?

Linear dependence and linear independence of solution Easy
A. The derivative of the other
B. A constant multiple of the other
C. The integral of the other
D. The square of the other

8 Which pair of functions is linearly independent?

Linear dependence and linear independence of solution Easy
A. and
B. and
C. and
D. and

9 The Wronskian is commonly used to test the linear dependence of which functions?

Linear dependence and linear independence of solution Easy
A. A set of constants
B. A set of variables
C. A set of functions
D. A set of equations

10 For two functions, the Wronskian is written as which expression?

Linear dependence and linear independence of solution Easy
A.
B.
C.
D.

11 In differential operator notation, what does represent?

Method of solution of linear differential equation using differential operator Easy
A.
B.
C.
D.

12 How is the equation written using ?

Method of solution of linear differential equation using differential operator Easy
A.
B.
C.
D.

13 What is the auxiliary equation of ?

Method of solution of linear differential equation using differential operator Easy
A.
B.
C.
D.

14 The complementary function of a homogeneous differential equation is obtained from its what?

Method of solution of linear differential equation using differential operator Easy
A. Particular integral
B. Auxiliary equation
C. Initial condition
D. Boundary value

15 For , what are the roots of the auxiliary equation?

Solution of second order homogeneous linear differential equation with constant coefficient Easy
A.
B.
C.
D.

16 If the auxiliary equation has two distinct real roots and , what is the complementary function?

Solution of second order homogeneous linear differential equation with constant coefficient Easy
A.
B.
C.
D.

17 If the repeated root of the auxiliary equation is , what is the complementary function?

Solution of second order homogeneous linear differential equation with constant coefficient Easy
A.
B.
C.
D.

18 If the roots are , what is the real complementary function?

Solution of second order homogeneous linear differential equation with constant coefficient Easy
A.
B.
C.
D.

19 For a third-order homogeneous linear differential equation, how many arbitrary constants usually appear in the general solution?

Solution of higher order homogeneous linear differential equations with constant coefficient Easy
A. Four
B. Three
C. One
D. Two

20 For , which is the auxiliary equation?

Solution of higher order homogeneous linear differential equations with constant coefficient Easy
A.
B.
C.
D.

21 Which of the following is a linear differential equation in ?

Introduction to linear differential equation Medium
A.
B.
C.
D.

22 What is the order and degree of the differential equation ?

Introduction to linear differential equation Medium
A. Order , degree
B. Order , degree
C. Order , degree
D. Order , degree

23 Solve .

Solution of linear differential equation Medium
A.
B.
C.
D.

24 For , find the general solution of .

Solution of linear differential equation Medium
A.
B.
C.
D.

25 The solution of satisfying is:

Solution of linear differential equation Medium
A.
B.
C.
D.

26 Using the Wronskian, determine whether and are linearly independent.

Linear dependence and linear independence of solution Medium
A. Independent because the Wronskian is
B. Dependent because one function contains
C. Independent because the Wronskian is
D. Dependent because the Wronskian is zero

27 Which set of functions is linearly independent on every interval?

Linear dependence and linear independence of solution Medium
A.
B.
C.
D.

28 If two solutions of a second-order homogeneous linear differential equation have a nonzero Wronskian at one point, what follows?

Linear dependence and linear independence of solution Medium
A. They are linearly dependent
B. They are linearly independent
C. They are necessarily identical
D. They must both be constant

29 In operator notation, the equation can be written as:

Method of solution of linear differential equation using differential operator Medium
A.
B.
C.
D.

30 If , then the operator applied to gives:

Method of solution of linear differential equation using differential operator Medium
A.
B.
C.
D.

31 The complementary function of is:

Method of solution of linear differential equation using differential operator Medium
A.
B.
C.
D.

32 The general solution of is:

Solution of second order homogeneous linear differential equation with constant coefficient Medium
A.
B.
C.
D.

33 Solve .

Solution of second order homogeneous linear differential equation with constant coefficient Medium
A.
B.
C.
D.

34 For , with and , the particular solution is:

Solution of second order homogeneous linear differential equation with constant coefficient Medium
A.
B.
C.
D.

35 Which differential equation has the complementary function ?

Solution of second order homogeneous linear differential equation with constant coefficient Medium
A.
B.
C.
D.

36 The general solution of is:

Solution of higher order homogeneous linear differential equations with constant coefficient Medium
A.
B.
C.
D.

37 The complementary function of is:

Solution of higher order homogeneous linear differential equations with constant coefficient Medium
A.
B.
C.
D.

38 The general solution of is:

Solution of higher order homogeneous linear differential equations with constant coefficient Medium
A.
B.
C.
D.

39 For , how many arbitrary constants appear in the complementary function?

Solution of higher order homogeneous linear differential equations with constant coefficient Medium
A. Six
B. Four
C. Five
D. Three

40 The general solution of is:

Solution of higher order homogeneous linear differential equations with constant coefficient Medium
A.
B.
C.
D.

41 Which of the following is a linear differential equation in , even though its coefficients are variable?

Introduction to linear differential equation Hard
A.
B.
C.
D.

42 For which value of is the equation not defined as a linear differential equation on the entire interval ?

Introduction to linear differential equation Hard
A.
B.
C.
D.

43 Find the general solution of on an interval not containing .

Solution of linear differential equation Hard
A.
B.
C.
D.

44 The initial-value problem , , has a unique solution on an interval provided which condition holds?

Solution of linear differential equation Hard
A. and are continuous on
B. is differentiable and is continuous on
C. is bounded and is differentiable on
D. is continuous on and is bounded on

45 Solve subject to .

Solution of linear differential equation Hard
A.
B.
C.
D.

46 On an interval containing , which pair of functions is linearly independent?

Linear dependence and linear independence of solution Hard
A.
B.
C.
D.

47 Let and be solutions of a second-order homogeneous linear equation on an interval . If their Wronskian satisfies at one point , what follows under the usual continuity assumptions?

Linear dependence and linear independence of solution Hard
A. They are linearly dependent on
B. Their equation must have constant coefficients
C. They are independent only at
D. They become independent away from

48 For and , which statement about their Wronskian is correct?

Linear dependence and linear independence of solution Hard
A. for all
B. for all
C. for all
D. for all

49 If , which factorization is correct for the operator ?

Method of solution of linear differential equation using differential operator Hard
A.
B.
C.
D.

50 For the equation , which particular integral is valid?

Method of solution of linear differential equation using differential operator Hard
A.
B.
C.
D.

51 Using operator notation, determine a particular integral for .

Method of solution of linear differential equation using differential operator Hard
A.
B.
C.
D.

52 For , which set forms the complete complementary solution?

Method of solution of linear differential equation using differential operator Hard
A.
B.
C.
D.

53 Find the general solution of .

Solution of second order homogeneous linear differential equation with constant coefficient Hard
A.
B.
C.
D.

54 The solution of satisfying and is:

Solution of second order homogeneous linear differential equation with constant coefficient Hard
A.
B.
C.
D.

55 Which equation has a general solution containing both exponential decay and oscillation with angular frequency ?

Solution of second order homogeneous linear differential equation with constant coefficient Hard
A.
B.
C.
D.

56 For , the roots are and . What are and ?

Solution of second order homogeneous linear differential equation with constant coefficient Hard
A.
B.
C.
D.

57 Find the complementary solution of .

Solution of higher order homogeneous linear differential equations with constant coefficient Hard
A.
B.
C.
D.

58 Determine the real general solution of .

Solution of higher order homogeneous linear differential equations with constant coefficient Hard
A.
B.
C.
D.

59 Which is the correct real general solution of ?

Solution of higher order homogeneous linear differential equations with constant coefficient Hard
A.
B.
C.
D.

60 For the characteristic roots with multiplicity three and each with multiplicity two, what is the order of the differential equation and how many arbitrary constants occur?

Solution of higher order homogeneous linear differential equations with constant coefficient Hard
A. Order , with arbitrary constants
B. Order , with arbitrary constants
C. Order , with arbitrary constants
D. Order , with arbitrary constants
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