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Quiz about Introduction to Differential Equations

Introduction to Differential Equations Quiz

This quiz covers some basic terms and classifications of differential equations. Everyone can have a try. Best of luck and enjoy!

A multiple-choice quiz by Matthew_07. Estimated time: 6 mins.

Author
Matthew_07

Time
6 mins

Type
Multiple Choice

Quiz #
278,779

Updated
Dec 03 21

# Qns
10

Difficulty
Tough

Avg Score
6 / 10

Plays
850

Last 3 plays: Guest 113 (9/10), Guest 2 (0/10), Guest 128 (5/10).

One at a Time - Single Page - Timed Game

Question 1 of 10

1. A mathematical equation is an equation where there are two equal terms on the left and right hand sides, connected by an equal sign, "=". For example, x + 2 = 5. On the other hand, there is another type of equation, namely the differential equation. Which of the following is a differential equation? Hint

sin x + cos y = tan z

3x^3 + 2x^2 + x + 1 = 0

x^2 + y^2 + z^2 = 1

(dy/dx)^2 + (d2y/dx2) = x + 2y

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Question 2 of 10

2. The term dy/dx in a differential equation is also known as the ____ of change of y with respect to x. Hint

Delta

Difference

Power

Rate

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Question 3 of 10

3. One way of classifying differential equations is by their orders. The order of a differential equation is the highest order of derivative that occurs in it. Which of the following is a third order differential equation? Hint

(d5y/dx5)^2 + (d2y/dx2)^3 = 0

(d3y/dx3)^2 + (d4y/dx4)^3 = 0

(d3y/dx3)^2 + (d2y/dx2)^3 = 0

(dy/dx)^2 + (d2y/dx2)^3 = 0

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Question 4 of 10

4. Another way of classifying differential equation is by their degrees. The degree of a differential equation is the degree (or power) of the highest order derivative. What is the degree of the following differential equation?
(d3y/dx3)^4 + (d2y/dx2)^5 = 0
Hint

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Question 5 of 10

5. A very common way to classify differential equations is by categorizing them into either linear differential equations or non-linear equations. There are 2 conditions a linear differential equation must fulfill. The first one is that all the degrees of y or any derivatives of y (y', y', y'', etc.) must be 1. The second one is that all the coefficients of y or any derivatives of y must be constants or functions of x. Is the differential equation (1+y)(dy/dx) + 2(d2y/dx2) + (x+3)(d3y/dx3)= 0 an linear one or a non-linear?

Answer: (Type L for linear or N for non-linear.)

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Question 6 of 10

6. Another way to categorize differential equations is by classifying them into either ordinary differential equations (ODE) or partial differential equations (PDE). Which of the following is the characteristic of a PDE? Hint

It has only one dependent variable.

It has more than one dependent variable.

It has more than one independent variable.

It has only one independent variable.

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Question 7 of 10

7. There are four types of first order differential equations, namely separable, homogeneous, exact and linear. The one that can be expressed in the form of M(x) dx + N(y) dy = 0 is classified as a (an) ____ first order differential equation. Hint

Homogeneous

Separable

Exact

Linear

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Question 8 of 10

8. The existence (the solution of a given differential equation exists) and uniqueness (there is only one solution for a given differential equation) of differential equations are well studied by mathematicians. Which of the following theorems is the one that defines the uniqueness of a given differential equation? Hint

Newton's Theorem

Taylor's Theorem

Picard's Theorem

Euler's Theorem

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Question 9 of 10

9. We will now try to solve a simple differential equation. If we integrate both sides of the differential equation dy/dx = x, we will get y = (x^2)/2 + c. However, if the given differential equation is of the form dy/dx = y, we will have to gather all the same terms at the same sides, namely dy/y = dx. Now, if we integrate both sides, what is the next equation we will obtain? Hint

In y = x + c

y = 1 + c

In y = 1 + c

y = x + c

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Question 10 of 10

10. Differential equations are used widely in the fields of economics, ecology and physics.

True

False

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Quiz Answer Key and Fun Facts

Answer: (dy/dx)^2 + (d2y/dx2) = x + 2y

As implied by its name, there must be at least one derivative in a differential equation. For instance, in the differential equation (dy/dx)^2 + (d2y/dx2) = x + 2y, the derivatives are dy/dx and also d2y/dx2.

Meanwhile, sin x + cos y = tan z is a trigonometric equation involving three unknowns, whereas x^2 + y^2 + z^2 = 1 is an equation describing a unit sphere. Last but not least, 3x^3 + 2x^2 + x + 1 = 0 is a polynomial equation of degree 3. (The highest power of this equation is 3)

2. The term dy/dx in a differential equation is also known as the ____ of change of y with respect to x.

Answer: Rate

Differential equations have wide applications in many fields, such as economics, ecology and physics. In ecology, the rate of change of a certain species' population can be studied by applying differential equation.

By solving the differential equation, we will eventually get an equation describing the population at a certain time, thus enabling us to predict the population in the future.

Answer: (d3y/dx3)^2 + (d2y/dx2)^3 = 0

There are 2 derivatives in the differential equation (d3y/dx3)^2 + (d2y/dx2)^3 = 0, namely d3y/dx3 (which is a third order) and d2y/dx2 (which is a second order). The highest order is therefore three. So, this is a third order differential equation.

Meanwhile, (d3y/dx3)^2 + (d4y/dx4)^3 = 0 is a fourth order differential equation. (dy/dx)^2 + (d2y/dx2)^3 = 0 and (d5y/dx5)^2 + (d2y/dx2)^3 = 0 are second and fifth order differential equations respectively.

Answer: 4

Notice that the highest order derivative in the differential equation is d3y/dx3. This term is raised to the power of 4. Therefore, this differential equation is of a fourth degree.

Notice that the other derivative (d2y/dx2) has a higher power (degree) of 5. However, keep in mind that when it comes to determinig the degree of a differential equation, we choose the derivative of the highest order. Clearly, the term d2y/dx2 is only a second order derivative, as opposed to the derivative d3y/dx3, which is a third order one.

Answer: N

Notice that all the derivatives dy/dx, d2y/dx2 and d3y/dx3 are of degree one. So, it is fine. The coefficient of d2y/dx2 is 2, which is a constant. On the other hand, the coefficient of d3y/dx3 is x+3, which is a function of x, which is fine also. However, the coefficient of dy/dx is 1+y, which is a function of y and NOT a function of x. So, it is a non-linear differential equation.

Answer: It has more than one independent variable.

For example, the differential equation dy/dx + dy/dz = 4 is a partial differential equation, since it has 2 independent variables, namely x and z. On the contrary, the differential equation dy/dx = 2 is an ordinary differential equation because it has only one independent variable, namely x.

Answer: Separable

Observe that M(x) dx + N(y) dy = 0 can be rewritten as M(x) dx = -N(y) dy. Then, we can solve the differential equation by integrating both sides.

Answer: Picard's Theorem

The fundamental criteria for a differential equation is to have a unique (only one) solution as given by the Picard's Theorem, where it states that we have to test for the continuity of the given function. For example, dy/dx = 3x. This equation is continuous on the xy-plane.

However, if the given differential equation is dy/dx = 1/x, then we should notice that 1/x is discontinuous at x = 0. Therefore, we conclude that the differential equation dy/dx = 1/x does not have a unique solution.

Answer: In y = x + c

If we integrate the term dy/y, we will get In y. On the right hand side, if we integrate dx, or 1 dx, we will obtain x + c. (Don't forget to add the arbitrary constant).

Now, if we continue the steps to solve for y in terms of x, we will get the following:
=> dy/dx = y
=> dy/y = dx
=> In y = x + c
=> y = e^(x+c)
=> y = e^x + e^c
Substituting another constant A = e^c, we will get the solution of y = Ae^x.

10. Differential equations are used widely in the fields of economics, ecology and physics.

Answer: True

In economics, differential equations are used in the calculation of compound interests. Meanwhile, the calculations of electric currents in circuits in the field of physics also require differential equations.

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I hope you enjoyed this quiz and learned something as well. Thanks for playing and have a nice day!

Source: Author Matthew_07

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