Introduction to Differential Equations Online Quiz | Sci / Tech

Quiz Answer Key and Fun Facts

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?

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.

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?

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.

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

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.

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: 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.

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?

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.

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.

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.

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?

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.

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?

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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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!

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