Video: Ordinary Differential Equations

Which of the following relationships is an ordinary differential equation? [A] 𝑧 = 5π‘₯𝑦 [B] d𝑦/dπ‘₯ + 𝑦 = 0 [C] πœ•Β²π‘§/πœ•π‘₯Β² = 0 [D] 𝑦 = √(π‘₯Β² βˆ’ 4)

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Video Transcript

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 π‘₯𝑦, we see that it contains no derivatives. And therefore, this is not a differential equation. It’s simply an equation relating the three variables π‘₯, 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 π‘₯ 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 π‘₯ 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 π‘₯. 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 π‘₯.

What this means is that the function 𝑧 is not just a function of π‘₯, but also of one or more other variables, such as 𝑦. The partial derivative of 𝑧 with respect to π‘₯ 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 π‘₯ 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.

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