Master Homogeneous Linear Equations: A Comprehensive Guide for Differential Equations

Question 1

For the equation y'' + 4y' + 3y = 0, what are the roots of the characteristic equation?

Accepted Answer

-1, -3

Answer

1, -3

Answer

-1, 3

Answer

1, 3

Question 2

Which of the following is the general solution to the equation y'' - 4y' + 4y = 0?

Accepted Answer

y = e^(2x)(c₁cos2x + c₂sin2x)

Answer

y = e^(2x)(c₁sin2x + c₂cos2x)

Answer

y = e^(-2x)(c₁cos2x + c₂sin2x)

Answer

y = e^(-2x)(c₁sin2x + c₂cos2x)

Question 3

What is the method used to solve homogeneous linear equations with constant coefficients called?

Accepted Answer

Characteristic equation method

Answer

Method of undetermined coefficients

Answer

Method of variation of parameters

Answer

Laplace transform method

Question 4

Which of the following is NOT a step in solving homogeneous linear equations using the characteristic equation method?

Accepted Answer

Solve the resulting differential equation using integration.

Answer

Determine the characteristic equation.

Answer

Find the roots of the characteristic equation.

Answer

Use the roots to construct the general solution.

Question 5

For the equation y''' - 2y'' + y' - 2y = 0, what is the order of the equation?

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 6

Which of the following is the characteristic equation for y'' + 2y' + 5y = 0?

Accepted Answer

r^2 + 2r + 5 = 0

Answer

r^2 - 2r + 5 = 0

Answer

r^2 + 2r - 5 = 0

Answer

r^2 - 2r - 5 = 0

Question 7

For the non-homogeneous equation y'' - 4y' + 3y = e^x, which method should be used to solve it?

Accepted Answer

Method of undetermined coefficients

Answer

Method of variation of parameters

Answer

Characteristic equation method

Answer

Laplace transform method

Question 8

Which method can be used to solve the homogeneous linear equation y'' + p(x)y' + q(x)y = 0 when the roots of the characteristic equation are complex conjugates?

Accepted Answer

Method of variation of parameters

Answer

Method of undetermined coefficients

Answer

Euler's method

Answer

Runge-Kutta method

Question 9

The homogeneous linear equation y'' - 2y' + 5y = 0 has characteristic equation r^2 - 2r + 5 = 0. What is the nature of its roots?

Accepted Answer

Complex conjugates

Answer

Real and distinct

Answer

Real and equal

Answer

No real roots

Question 10

Consider the differential equation y'' - 4y = 0. If r₁ = 2 is a root of the characteristic equation, what is the complementary solution?

Accepted Answer

y = C₁e^2x + C₂e^(-2x)

Answer

y = C₁e^2x - C₂e^(-2x)

Answer

y = C₁(e^2x + e^(-2x)) + C₂(e^2x - e^(-2x))

Answer

y = C₁(e^2x - e^(-2x)) + C₂(e^2x + e^(-2x))

Question 11

Solve the differential equation y'' + 4y' + 4y = 0. Determine the general solution.

Accepted Answer

y = C₁cos(2x) + C₂sin(2x)

Answer

y = C₁e^x + C₂xe^x

Answer

y = C₁e^(-x) + C₂xe^(-x)

Answer

y = C₁cos(2x) + C₂sin(2x) + C₃xcos(2x) + C₄xsin(2x)

Question 12

Solve the differential equation y'' - 2y' + 2y = 0. Find the general solution.

Accepted Answer

y = C₁e^x + C₂e^(-x)

Answer

y = C₁e^x - C₂e^(-x)

Answer

y = C₁(e^x + e^(-x)) + C₂(e^x - e^(-x))

Answer

y = C₁(e^x - e^(-x)) + C₂(e^x + e^(-x))

Question 13

Consider the differential equation y''' - 3y'' + 2y' = 0. What is the order of this equation?

Accepted Answer

Third-order

Answer

First-order

Answer

Second-order

Answer

Fourth-order

Question 14

Solve the differential equation y'' + 9y = 0. Determine the nature of the solution.

Accepted Answer

Exponential solution

Answer

Trigonometric solution

Answer

Hyperbolic solution

Answer

Polynomial solution

Question 15

How do the roots of the characteristic polynomial of a homogeneous linear equation determine the behavior of its solutions?

Accepted Answer

The roots determine the number and types of solutions.

Answer

The degree of the polynomial equals the order of the equation.

Answer

The coefficients of the polynomial match those of the original equation.

Answer

The polynomial has no relation to the solutions.

Question 16

Consider the homogeneous linear equation y'' + p(x)y' + q(x)y = 0, where y1(x) is a solution. The second linearly independent solution can be found using:

Accepted Answer

y2(x) = y1(x) ∫ (1/y1(x)^2) dx

Answer

y2(x) = y1(x) ∫ (y1(x)^2) dx

Answer

y2(x) = ∫ y1(x) dx

Answer

The second solution cannot be determined.

Question 17

Regarding the characteristic equation of a homogeneous linear equation of order n, which statement is true?

Accepted Answer

It is a polynomial equation of degree n.

Answer

It is a polynomial equation of degree 2n.

Answer

It is a transcendental equation.

Answer

It is an algebraic equation of degree n.

Question 18

Determine if the equation y'' + 2xy' + (x^2 - 1)y = 0 is homogeneous:

Accepted Answer

No, the equation is not homogeneous.

Answer

Yes, the equation is homogeneous.

Answer

Cannot be determined.

Question 19

The characteristic equation for the differential equation y' - 3y = 0 is:

Accepted Answer

r - 3 = 0

Answer

r + 3 = 0

Answer

r^2 - 3 = 0

Answer

r^2 + 3 = 0

Question 20

For the differential equation y'' + 4y = 0, the general solution is:

Accepted Answer

y = c1cos(2x) + c2sin(2x)

Answer

y = c1e^(2x) + c2e^(-2x)

Answer

y = c1x + c2

Answer

y = c1e^(4x) + c2xe^(4x)

Question 21

If the roots of the characteristic equation are r₁ = 2 and r₂ = -3, the general solution to the corresponding homogeneous linear equation is:

Accepted Answer

y = c₁e^(2x) + c₂e^(-3x)

Answer

y = c₁cos(2x) + c₂sin(2x)

Answer

y = c₁e^(2x) + c₂cos(3x)

Answer

y = c₁sin(2x) + c₂cos(3x)

Question 22

Which of the following is a necessary condition for a system of homogeneous linear equations to have a non-trivial solution?

Accepted Answer

The determinant of the coefficient matrix is zero.

Answer

The system has more equations than variables.

Answer

The system has more variables than equations.

Answer

The coefficient matrix is invertible.

Question 23

Consider the system of homogeneous linear equations:
x + 2y = 0
3x - y = 0
      The general solution to this system is:

Accepted Answer

x = c₁, y = -2c₁

Answer

x = -2c₁, y = c₁

Answer

x = c₁, y = c₂

Answer

x = c₁ + c₂, y = -2c₁ - 2c₂

Question 24

Which of the following is a limitation of the method of undetermined coefficients?

Accepted Answer

Can only be used for equations with constant coefficients.

Answer

Can only be used for equations with exponential forcing functions.

Answer

Can only be used for equations with polynomial forcing functions.

Answer

Can only be used for equations with trigonometric forcing functions.

Question 25

Consider the differential equation: y'' - 5y' + 6y = 0. The characteristic equation of this equation is:

Accepted Answer

r^2 - 5r + 6 = 0

Answer

r^2 + 5r + 6 = 0

Answer

r^2 - 6r + 5 = 0

Answer

r^2 + 6r + 5 = 0

Question 26

Consider the differential equation y''' + 2y'' - 3y' = 0. The characteristic equation of this equation is:

Accepted Answer

r^3 + 2r^2 - 3r = 0

Answer

r^3 - 2r^2 - 3r = 0

Answer

r^3 - 2r^2 + 3r = 0

Answer

r^3 + 2r^2 + 3r = 0

Question 27

Which of the following is a second-order homogeneous linear equation:

Accepted Answer

y'' + 2y' + y = 0

Answer

y' + 2y = 0

Answer

y''' - 2y' + y = 0

Answer

y'' + 2y + y' = 0

Question 28

Find the general solution to the differential equation y'' + 4y = 0. (The characteristic equation is r^2 + 4 = 0.)

Accepted Answer

y(x) = C₁cos(2x) + C₂sin(2x)

Answer

y(x) = C₁ + C₂x

Answer

y(x) = C₁xcos(2x) + C₂xsin(2x)

Answer

y(x) = C₁e^(2x) + C₂e^(-2x)

Question 29

Which of the following statements is true about the solutions to the differential equation y'' + 2y' + y = 0?

Accepted Answer

The solutions are linearly independent.

Answer

The solutions are decreasing.

Answer

The solutions are linearly dependent.

Answer

The solutions are periodic.

Question 30

Find the general solution to the system of differential equations:
x' = x + 2y,
y' = 2x + y.

Accepted Answer

x(t) = c₁(e^(-t) + e^(2t)) + c₂e^(t), y(t) = c₁(e^(-t) - e^(2t)) + c₂e^(t)

Answer

x(t) = c₁(e^(-t) + e^(2t)), y(t) = c₁(e^(-t) + e^(2t))

Answer

x(t) = c₁e^(-t) + c₂e^(2t), y(t) = c₁e^(-t) - c₂e^(2t)

Question 31

Determine the general solution to the homogeneous linear equation: y'' - 2y = 0.

Accepted Answer

y = C1e^(2x) + C2e^(-2x)

Answer

y = C1sin(2x) + C2cos(2x)

Answer

y = C1e^x + C2e^(-x)

Answer

y = C1x + C2

Question 32

Which concept is being tested by the following question: 'Find the values of r that satisfy the characteristic equation r^3 - 2r^2 + r - 2 = 0.'?

Accepted Answer

Characteristic roots

Answer

Linear independence

Answer

Variation of parameters

Answer

Wronskian

Question 33

Solve the homogeneous linear equation: y''' - 3y'' + 3y' - y = 0.

Accepted Answer

y = C1e^x + C2xe^x + C3x^2e^x

Answer

y = C1e^-x + C2xe^-x + C3x^2e^-x

Answer

y = C1 + C2x + C3x^2

Answer

y = C1sin(x) + C2cos(x) + C3x

Question 34

Determine the truthfulness of the statement: 'If the characteristic roots of a homogeneous linear equation are distinct, then the equation has a set of linearly independent solutions.'

Accepted Answer

True

Answer

It depends on the coefficients of the equation

Answer

False

Answer

Only true if the roots are real

Question 35

Identify the homogeneous linear equation with constant coefficients among the following:

Accepted Answer

y'' + 4y' + 3y = 0

Answer

y' + xy = 2

Answer

y'' + y^2 = 0

Answer

y' + sin(x)y = cos(x)

Question 36

Determine the characteristic equation for the homogeneous linear equation y'' + 2y' - 3y = 0.

Accepted Answer

r^2 + 2r - 3 = 0

Answer

r^2 + 2r + 3 = 0

Answer

r^2 - 2r - 3 = 0

Question 37

If the characteristic equation of a homogeneous linear equation is r^2 - 4 = 0, express the general solution of the differential equation.

Accepted Answer

y = c1*e^(2x) + c2*e^(-2x)

Answer

y = c1*e^(4x) + c2*e^(-4x)

Answer

y = c1*e^(2x) + c2*xe^(2x)

Answer

y = c1*cos(2x) + c2*sin(2x)

Question 38

Find the general solution to the homogeneous linear equation y'' + 6y' + 9y = 0.

Accepted Answer

y = (c1 + c2*x)*e^(-3x)

Answer

y = c1*cos(3x) + c2*sin(3x)

Answer

y = c1*e^(-3x) + c2*e^(3x)

Answer

y = (c1 + c2*x)*e^(3x)

Question 39

For the characteristic equation of a homogeneous linear equation r^2 + 4r + 13 = 0, determine the form of the general solution.

Accepted Answer

y = e^(-2x)*(c1*cos(3x) + c2*sin(3x))

Answer

y = (c1 + c2*x)*e^(-2x)

Answer

y = c1*e^(2x) + c2*e^(-2x)

Answer

y = c1*cos(3x) + c2*sin(3x)

Question 40

Which statement is incorrect regarding homogeneous linear equations with constant coefficients?

Accepted Answer

The solution is always a linear combination of exponential functions.

Answer

The equation is linear in the dependent variable and its derivatives.

Answer

The coefficients of the dependent variable and its derivatives are constants.

Question 41

A spring-mass system with no damping force is modeled by the equation m*y'' + ky = 0, where m is the mass and k is the spring constant. What type of equation is this?

Accepted Answer

Homogeneous linear equation with constant coefficients

Answer

Homogeneous nonlinear equation

Answer

Non-homogeneous linear equation with constant coefficients

Question 42

If the roots of the characteristic equation of a homogeneous linear equation are purely imaginary, what can be said about the solutions of the differential equation?

Accepted Answer

The solutions are periodic functions.

Answer

The solutions are exponentially decaying functions.

Answer

The solutions are linear functions.

Answer

The solutions are exponentially growing functions.

Question 43

The Wronskian of two solutions to a second-order homogeneous linear equation is found to be zero for all x. What does this imply about the solutions?

Accepted Answer

The solutions are linearly dependent.

Answer

The solutions are periodic.

Answer

The solutions are orthogonal.

Answer

The solutions are linearly independent.