Video: Basics of Differential Equations

In this video, we will learn how to classify differential equations into ordinary or partial and linear or nonlinear and identify the order of the differential equations.

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Video Transcript

In this video, we will learn about the basics of differential equations. We’ll see first what a differential equation is and how it might arise in physical problems within the areas of mathematics and physics. We’ll introduce the terminology associated with differential equations. And in the context of examples, we’ll see how to classify differential equations according to their type and order.

Let’s begin then by discussing what a differential equation actually is. Well, it’s just an equation that contains a function and one or more of its derivatives with respect to an independent variable. For example, the equation d𝑦 by d𝑥 plus 𝑦 equals three 𝑥 is a differential equation. Here, 𝑦 is the function we’re interested in. It’s a function of the independent variable 𝑥 and the equation includes the first derivative of 𝑦 with respect to 𝑥.

Far more complicated differential equations also exist. For example, the equation d two 𝑦 by d𝑥 squared plus four 𝑦 multiplied by d𝑦 by d𝑥 all squared plus three 𝑦 equals sin 𝑥. This differential equation includes not only a first derivative d𝑦 by d𝑥 which is squared, but it also includes the second derivative of 𝑦 with respect to 𝑥, d two 𝑦 by d𝑥 squared. So, we have an equation that contains a function 𝑦, and its first derivative d𝑦 by d𝑥, and also its second derivative d two 𝑦 by d𝑥 squared.

The first of these two examples is known as a first order differential equation because the highest order derivative it contains is a first derivative, the first derivative of 𝑦 with respect to 𝑥, d𝑦 by d𝑥. The second example is known as a second order differential equation because the highest order derivative it contains is a second derivative, d two 𝑦 by d𝑥 squared. More generally, differential equations can be classified according to the highest order derivative they contain. So, a differential equation in which the highest order derivative is an 𝑛th derivative would be classed as an 𝑛th order differential equation.

Examples of differential equations arise in many physical problems. For example, simple population growth, where the rate of increase of the population 𝑃 is proportional to the population itself. And this can be modeled by the differential equation d𝑃 by d𝑡 equals 𝑘𝑃. Here, 𝑡 represents time, and 𝑘 is called the constant of proportionality.

Now, it’s beyond the scope of this video to see how we would solve a differential equation of this type. But in fact, if we had the initial condition that the population at time zero is equal to some value 𝑃 nought, then it can be shown that the solution to this differential equation would be that the population at time 𝑡 is equal to 𝑃 nought multiplied by 𝑒 to the power of 𝑘𝑡. So, the population grows in an exponential way.

Now, to solve a differential equation in general, we need to find an expression for the dependent variable, which in this case would be 𝑃, in terms of the independent variable, which here would be 𝑡. The solution is a function such that that function and its derivatives satisfy the differential equation. Now, generally, when we solve a differential equation, we get not a single solution but a family of solutions, all of which will satisfy the differential equation but will differ from one another in the values of any constants.

If we’re also provided with additional information, such as here we were given the value of the population at time zero, then we can determine the values of these constants and hence obtain a particular solution to the differential equation. This additional information is known as a boundary condition, or sometimes an initial condition if it specifies the value of the function when the value of the independent variable is zero.

However, solving differential equations is not the focus of this particular video. Rather, we are focusing on understanding and classifying them. We will, however, consider an example of how we can confirm that a given function does indeed satisfy a differential equation.

Is the function 𝑦 equals one over two plus 𝑥 a solution to the differential equation 𝑦 prime equals negative 𝑦 squared?

Remember that 𝑦 prime is another way of saying d𝑦 by d𝑥, the first derivative of 𝑦 with respect to 𝑥. So, we’ve been given a first order differential equation, and we want to know whether the given function 𝑦 is a solution to it. That is, we need to know whether the function 𝑦 satisfies this equation.

Let’s begin then by first working out what 𝑦 prime, or d𝑦 by d𝑥, is equal to for this function 𝑦. And to do this, we can first express 𝑦 in an alternative form. We can write it as two plus 𝑥 to the power of negative one. We can then find this derivative using the general power rule, which says that if we have some function 𝑓 of 𝑥 to the power of 𝑛, then its derivative with respect to 𝑥 is equal to 𝑛 multiplied by 𝑓 prime of 𝑥 multiplied by 𝑓 of 𝑥 to the power of 𝑛 minus one.

Here, our function 𝑓 of 𝑥 is two plus 𝑥, and our power 𝑛 is negative one. So, applying the general power rule, we have 𝑛, that’s negative one, multiplied by the derivative of two plus 𝑥, which is just one, multiplied by 𝑓 of 𝑥. That’s two plus 𝑥, to the power of 𝑛 minus one. So, that’s the power of negative two. We can then rewrite this as negative one over two plus 𝑥 all squared. So, we know what the left-hand side of this differential equation would be for this function 𝑦.

On the right-hand side, we have negative 𝑦 squared. So, that’s the original function 𝑦 squared and then multiply it by negative one, which is equal to negative one over two plus 𝑥 all squared. To square a fraction, we can square the numerator and square the denominator. So, we have negative one squared, which is one, over two plus 𝑥 all squared.

Now, we compare our expressions for 𝑦 prime and negative 𝑦 squared. And we see that they are both equal to negative one over two plus 𝑥 all squared. And therefore, they are indeed equal to one another. This tells us that the function 𝑦 equals one over two plus 𝑥 does satisfy the given differential equation and, therefore, it is a solution.

We’ll now look at some further examples to introduce the terminology needed in order to describe the wide variety of other types of differential equations we may encounter.

Determine the order of the differential equation d two 𝑦 by d𝑥 squared cubed minus 𝑦 triple prime to the fourth power plus 𝑥 is equal to zero.

We recall first that the order of a differential equation is the order of the highest order derivative that appears in that equation. We can see at a glance that this differential equation involves a second derivative, d two 𝑦 by d𝑥 squared. But if we look a little closer, we see that the equation also contains 𝑦 triple prime, which is alternative notation for third derivative. The highest order derivative is three. And hence, the order of this differential equation is three.

Now, don’t be misled by the powers here. That is the power of three with the second derivative and the power of four with the third derivative. The order of a differential equation is not the highest power of the variable or any of its derivative that appears in the equation. It’s the order of the highest order derivative in the equation. So, that power of three for the first term and the power of four for the second term are entirely irrelevant in terms of determining the differential equation’s order.

In our next example, we’ll learn the difference between linear and nonlinear differential equations.

Is the differential equation d𝑦 by d𝑥 plus 𝑥 root 𝑦 equals 𝑥 squared linear?

A linear differential equation is one which can be expressed as a linear polynomial of the unknown function, in this case 𝑦 and its derivatives. What this means is that the only powers of the unknown function and each derivative that appear in the equation are one or zero if the equation does not contain that order derivative. And also each derivative and the function itself are multiplied by functions of 𝑥 only.

So, for example, the equation two times d𝑦 by d𝑥 plus four 𝑥𝑦 equals three 𝑥 would be an example of a linear differential equation. Because the power of both 𝑦 and d𝑦 by d𝑥 is one, and they’re each multiplied by a function of 𝑥 only. Whereas the equation two times d𝑦 by d𝑥 plus four 𝑥 over 𝑦 equals three 𝑥 is nonlinear as in the second term, the power of 𝑦 is negative one. The equation four 𝑥 d two 𝑦 by d𝑥 squared plus two 𝑦 d𝑦 by d𝑥 equals seven is also nonlinear as in the second term, we see that d𝑦 by d𝑥 is multiplied by a function of 𝑦, not a pure function of 𝑥.

More formally, we can say that a differential equation is linear if it can be expressed in the form shown on the screen. Each 𝑛th order derivative of 𝑦 and the function 𝑦 itself is multiplied by a polynomial in 𝑥 only. So, let’s consider the differential equation that we’ve been given. And we can see that it includes the square root of 𝑦. Now, another way of expressing the square root of 𝑦 is as 𝑦 to the power of one-half. And hence, this differential equation is nonlinear, as the power of 𝑦 is not equal to one.

Now, notice that it isn’t the presence of the 𝑥 squared term on the right-hand side that makes this differential equation nonlinear. 𝑥 is the independent variable in this equation. And it is only the powers of the dependent variable and its derivatives, that’s 𝑦 and d𝑦 by d𝑥 and so on, that must all be equal to one in order to make the equation linear.

In our final example, we’ll learn the difference between ordinary and partial differential equations.

Which of the following relationships is an ordinary differential equation?

Recall, first of all, that a differential equation contains a function and one or more of its derivatives with respect to an independent variable. If we consider the first equation, first of all, 𝑧 equals five 𝑥𝑦, we see that it contains no derivatives. And therefore, this is not a differential equation. It’s simply an equation relating the three variables 𝑥, 𝑦, and 𝑧. So, we can rule out option A.

In the same way, if we consider the final equation 𝑦 equals the square root of 𝑥 squared minus four, this isn’t a differential equation either, as it doesn’t contain any derivatives. It just expresses the relationship between the variables 𝑥 and 𝑦. So, we’re left with just two possibilities, B and C. Considering the second equation, we see that it contains an unknown variable 𝑦 and its derivative with respect to an independent variable 𝑥. So, this is an example of a differential equation.

But the question doesn’t just ask us for which is a differential equation. It asks us, which is an ordinary differential equation. So, we need to consider what this word ordinary means in this context. The third equation does also contain a derivative. And in fact, it is a second derivative this time. But we see that the notation used is slightly different. This notation represents the partial second derivative of the variable 𝑧 with respect to 𝑥.

What this means is that the function 𝑧 is not just a function of 𝑥, but also of one or more other variables, such as 𝑦. The partial derivative of 𝑧 with respect to 𝑥 is the function we get if we treat each of the other variables as constant when differentiating. And in fact, the partial second derivative of 𝑧 with respect to 𝑥 is what we get if we do this twice.

So, we return to that word ordinary in the question. An ordinary differential equation contains only ordinary, as opposed to partial derivatives, as the unknown function is a function of the independent variable only. From the notation used in equation B, we see that this contains only the function 𝑦 and its ordinary first derivative. So, this is an ordinary differential equation. Whereas option C contains a partial derivative, and so it is known as a partial differential equation.

You may see the abbreviations O.D.E and P.D.E used to describe ordinary and partial differential equations, respectively. So, our answer to the question, which of the following relationships is an ordinary differential equation, is B. A and D are not differential equations. And C is a differential equation, but it is a partial differential equation.

Let’s summarise then what we’ve seen in this video. Firstly, differential equations are equations which relate a function and one or more of its derivatives. The order of a differential equation is the order of the highest order derivative it contains. So, a differential equation in which the highest order derivative was a third derivative would have an order of three.

A linear differential equation can be expressed in the form shown onscreen. The power of the function 𝑦 and each of its derivatives is either one or zero. And 𝑦 and each of its derivatives are multiplied by pure functions of 𝑥. And finally, ordinary differential equations, or O.D.E.s, contain only ordinary derivatives, such as d𝑦 by d𝑥. Here, the function 𝑦 is a function of only one independent variable 𝑥.

Whereas partial differential equations, or P.D.E.s, contain partial derivatives. Here, the function 𝑦 is a function of more than one independent variable. Differential equations have many applications and can be used to model a wide range of physical phenomena.

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