1 $Which of the following defines a linear differential equation?$

A.

The dependent variable and its derivatives appear with degree 1 and are not multiplied together.

B.

The dependent variable appears with degree 1, but derivatives can be of higher degree.

C.

The derivatives are multiplied together.

D.

The independent variable appears with degree 1 only.

2 $Identify the non-linear differential equation from the following:$

A.

B.

C.

D.

3 $The general form of an -th order linear homogeneous differential equation with constant coefficients is:$

A.

B.

C.

D.

4 $If and are two solutions of a homogeneous linear differential equation, then by the Principle of Superposition, which of the following is also a solution?$

A.

B.

C.

D.

5 $Using the differential operator, the equation can be written as:$

A.

B.

C.

D.

6 $Two functions and are said to be linearly dependent on an interval if:$

A.

Their Wronskian for all .

B.

There exist constants not both zero such that for all .

C.

.

D.

They are solutions to different differential equations.

7 $The Wronskian of two functions and is defined by the determinant:$

A.

B.

C.

D.

8 $Calculate the Wronskian of the functions and .$

A.

$0$

B.

C.

$2$

D.

9 $If the Wronskian of a set of solutions is non-zero at a point in the interval of interest, the solutions are:$

A.

Linearly Dependent

B.

Linearly Independent

C.

Oscillatory

D.

Imaginary

10 $Which of the following pairs of functions are linearly independent?$

A.

and

B.

and

C.

and

D.

and

11 $The auxiliary equation (or characteristic equation) for the differential equation is:$

A.

B.

C.

D.

12 $If the roots of the auxiliary equation are real and distinct (), the general solution is:$

A.

B.

C.

D.

13 $If the roots of the auxiliary equation are real and equal (), the general solution is:$

A.

B.

C.

D.

14 $If the roots of the auxiliary equation are complex conjugate pairs, the general solution is:$

A.

B.

C.

D.

15 $Solve the differential equation: .$

A.

B.

C.

D.

16 $Solve the differential equation: .$

A.

B.

C.

D.

17 $Find the general solution of .$

A.

B.

C.

D.

18 $The roots of the auxiliary equation for are:$

A.

$1, 1$

B.

C.

D.

19 $For the differential equation, the roots of the auxiliary equation are:$

A.

B.

C.

$1, 2, 3$

D.

$0, 1, 6$

20 $The general solution of is:$

A.

B.

C.

D.

21 $Which differential equation has the solution ?$

A.

B.

C.

D.

22 $Solve .$

A.

B.

C.

D.

23 $What is the order of the differential equation whose auxiliary equation is ?$

A.

3

B.

4

C.

2

D.

1

24 $Given the roots of an auxiliary equation are $1, 1, 1$, the linearly independent solutions are:$

A.

B.

C.

D.

25 $The solution to represents:$

A.

Exponential growth

B.

Exponential decay

C.

Simple Harmonic Motion

D.

Linear motion

26 $Solve .$

A.

B.

C.

D.

27 $Which of the following functions generates the differential equation ?$

A.

B.

C.

D.

28 $If the auxiliary equation has roots (repeated twice), i.e.,, the solution is:$

A.

B.

C.

D.

29 $The number of arbitrary constants in the general solution of an -th order linear differential equation is:$

A.

B.

C.

D.

30 $For the equation, the general solution is:$

A.

B.

C.

D.

31 $The differential operator satisfies the law of indices :$

A.

B.

C.

D.

32 $Solve .$

A.

B.

C.

D.

33 $What is the nature of the roots for the DE ?$

A.

Complex conjugate

B.

Real and distinct

C.

Real and equal

D.

Rational

34 $If and, what is the differential equation satisfied by these functions?$

A.

B.

C.

D.

35 $Which of the following is the auxiliary equation for ?$

A.

B.

C.

D.

36 $Solve the equation from the previous question: .$

A.

B.

C.

D.

37 $The determinant condition is used to check:$

A.

Linear independence of solutions

B.

Continuity of solutions

C.

Differentiability of solutions

D.

Linearity of the equation

38 $Solve .$

A.

B.

C.

D.

39 $What substitution is used to convert into an algebraic equation?$

A.

B.

C.

D.

40 $If roots of AE are $0, 0$, the solution is:$

A.

B.

C.

D.

41 $Which of the following represents the differential operator for ?$

A.

B.

C.

D.

42 $For the equation, if, which term vanishes from the general solution ?$

A.

B.

C.

Both

D.

Neither

43 $The complementary function (C.F.) of a homogeneous linear differential equation is:$

A.

The general solution itself

B.

Zero

C.

Part of the particular integral

D.

Undefined

44 $Solve .$

A.

B.

C.

D.

45 $If the auxiliary equation is, the differential equation is:$

A.

B.

C.

D.

46 $Which of the following is NOT a property of the linear differential operator ?$

A.

B.

for constant

C.

D.

47 $Find the roots of the AE for .$

A.

B.

$2, 2, 2$

C.

D.

$0, 0, 8$

48 $The solution corresponds to the differential equation:$

A.

B.

C.

D.

49 $If the Wronskian everywhere on an interval, then and are:$

A.

Linearly Dependent

B.

Linearly Independent

C.

Orthogonal

D.

Reciprocal

50 $Solve the 4th order equation .$

A.

B.

C.

D.

Unit 2 - Practice Quiz

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