Ordinary Differential Equation

Homogeneous Equation

Quiz Type

Practice

Subject

Mathematics

Level

Level 300

Questions

10 questions

Created By

Isaac Baiden

Quick Rules

Time limit: 10 minutes
Multiple attempts are not allowed
All questions must be answered to submit

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Question 1 of 10

Question 1

Basic multiple choice Remembering

What is a homogeneous equation?

An equation where all terms are of the same degree

Correct

An equation with no solutions

An equation that has at least one variable

An equation that must include a constant

Explanation

A homogeneous equation is defined as an equation where all terms have the same degree, meaning each term is proportional to a certain degree of the variables involved.

Question 2

Basic multiple choice Remembering

Which of the following is an example of a homogeneous equation?

x^2 + y^2 = 0

Correct

x + y = 5

x^3 - 3x = 7

x^2 + 2 = y

Explanation

The equation x^2 + y^2 = 0 is homogeneous because both terms are of the same degree (2).

Question 3

Basic multiple choice Understanding

In the context of mathematical equations, what does 'homogeneous' refer to?

Uniformity in degree among terms

Correct

The presence of an inequality

A requirement for integer solutions

The use of multiple variables only

Explanation

'Homogeneous' refers to the property of having uniformity in the degree of terms within the equation.

Question 4

Basic multiple choice Understanding

What is a characteristic of a homogeneous linear equation?

It can be expressed in the form ax + by = 0

Correct

It always has two solutions

It must contain a constant term

It is not solvable

Explanation

A homogeneous linear equation can be expressed in the form ax + by = 0, where a and b are coefficients.

Question 5

Basic multiple choice Understanding

Which statement best describes the solutions to a homogeneous equation?

The solutions form a vector space

Correct

The only solution is always zero

They involve distinct real numbers

They are limited to integer values

Explanation

The solutions to a homogeneous equation form a vector space, which includes the trivial solution and potentially infinitely many other solutions.

Question 6

Complex multiple choice Understanding

Consider a homogeneous equation of the form ax + by = 0. If a = 3 and b = 6, which of the following statements correctly describes the relationship between the solutions of the equation and the concept of linear independence?

The solutions are linearly dependent if they can be expressed as scalar multiples of each other.

Correct

The solutions are independent because a homogeneous equation always has a nontrivial solution.

Since this is a homogeneous equation, all solutions must be equal to zero, which indicates linear independence.

The solutions will be linearly independent if a and b are both non-zero.

Explanation

The correct statement indicates that the solutions to the homogeneous equation ax + by = 0 depend on the ratios of the coefficients a and b. If solutions can be represented as scalar multiples, they are linearly dependent. The other statements misinterpret the implications of linear independence and trivial solutions in homogeneous equations.

Question 7

Complex multiple choice Understanding

A researcher is exploring a homogeneous equation represented as x - 2y = 0. If the researcher wants to find a non-trivial solution to this equation, which of the following scenarios best demonstrates that requirement?

Setting x = 2 and y = 1 satisfies the equation, indicating a non-trivial solution.

Correct

The only solution is x = 0 and y = 0, which is a trivial solution and thus cannot be used.

Choosing x = 1 and y = 1 gives a valid solution, but it actually leads to a different equation.

Selecting x = -2 and y = -1 would yield a non-trivial solution since they are non-zero values.

Explanation

The correct choice shows that the non-trivial solution arises from values that satisfy the original homogeneous equation without both variables being zero. The other options misunderstand the definition of trivial vs. non-trivial solutions or offer incorrect combinations of variables.

Question 8

Case based multiple choice Understanding

[Case Scenario] A mathematical problem involves a homogeneous equation, which is defined as an equation in which all terms are of the same degree. For example, the equation ax + by = 0 is a common example of a homogeneous linear equation. The equation implies a linear relationship between x and y, and it passes through the origin (0,0). Several students are given the task to analyze and graph this equation. Question: What characteristic should the students particularly note while working with the graph of a homogeneous equation like ax + by = 0?

The graph will always be a straight line that does not intersect the y-axis.

The graph will pass through the origin, indicating that there is no constant term.

Correct

The graph will represent a parabola, suggesting a second-degree relationship.

The graph will form a circle in the first quadrant because it is homogeneous.

Explanation

The critical aspect of homogeneous equations, especially linear ones such as ax + by = 0, is that they pass through the origin due to the absence of a constant term. This characteristic is vital for students to comprehend when analyzing their graphs.

Question 9

Case based multiple choice Analyzing

[Case Scenario] A group of advanced mathematics students is studying homogeneous equations. They are tasked with solving a homogeneous system of equations represented by the matrix form Ax = 0, where A is a coefficient matrix. One particular matrix they encounter is: A = \[ \begin{bmatrix} 2 & 3 \\ 4 & 6 \end{bmatrix} \]. They begin to analyze the implications of this system. Question: Based on their exploration of the matrix A, what conclusion can the students draw about the solutions of the homogeneous equation Ax = 0?

The system has only the trivial solution because the rows of the matrix are linearly independent.

The system has infinitely many solutions because the rows of the matrix A are linearly dependent.

Correct

The system has exactly two unique solutions due to the size of the matrix.

The system cannot be solved as the determinant of the matrix A is not defined.

Explanation

In analyzing the matrix A, students can conclude that the rows are linearly dependent, meaning that the homogeneous equation Ax = 0 will have infinitely many solutions, rather than just a trivial solution.

Question 10

Case based multiple choice Understanding

[Case Scenario] In a recent mathematics workshop, learners were introduced to homogeneous equations and their applications in various fields, including physics and engineering. They were presented with a physical problem involving forces acting in a system represented by a homogeneous equation. The equation provided was: F_x + F_y + F_z = 0. This equation indicates that the vector sum of forces in three dimensions equals zero. After discussing various force components, they were asked to evaluate the implications of this equation. Question: What can be inferred about the physical system if the equation F_x + F_y + F_z = 0 is satisfied?

The system is experiencing no net force, hence in equilibrium.

Correct

The system is accelerating in the direction of F_x.

The forces are only acting in the vertical direction with no horizontal components.

There are unbalanced forces present in the system, leading to motion.

Explanation

The equation F_x + F_y + F_z = 0 demonstrates that in a system where these forces are present, the sum of the forces results in no net force acting on the system, indicating that it is in a state of equilibrium.