WEBVTT
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Which of the following relationships is an ordinary differential equation?
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Recall, first of all, that a differential equation contains a function and one or more of its derivatives with respect to an independent variable.
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If we consider the first equation, first of all, π§ equals five π₯π¦, we see that it contains no derivatives.
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And therefore, this is not a differential equation.
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Itβs simply an equation relating the three variables π₯, π¦, and π§.
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So, we can rule out option A.
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In the same way, if we consider the final equation π¦ equals the square root of π₯ squared minus four, this isnβt a differential equation either, as it doesnβt contain any derivatives.
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It just expresses the relationship between the variables π₯ and π¦.
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So, weβre left with just two possibilities, B and C.
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Considering the second equation, we see that it contains an unknown variable π¦ and its derivative with respect to an independent variable π₯.
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So, this is an example of a differential equation.
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But the question doesnβt just ask us for which is a differential equation.
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It asks us, which is an ordinary differential equation.
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So, we need to consider what this word ordinary means in this context.
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The third equation does also contain a derivative.
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And in fact, it is a second derivative this time.
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But we see that the notation used is slightly different.
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This notation represents the partial second derivative of the variable π§ with respect to π₯.
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What this means is that the function π§ is not just a function of π₯, but also of one or more other variables, such as π¦.
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The partial derivative of π§ with respect to π₯ is the function we get if we treat each of the other variables as constant when differentiating.
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And in fact, the partial second derivative of π§ with respect to π₯ is what we get if we do this twice.
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So, we return to that word ordinary in the question.
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An ordinary differential equation contains only ordinary, as opposed to partial derivatives, as the unknown function is a function of the independent variable only.
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From the notation used in equation B, we see that this contains only the function π¦ and its ordinary first derivative.
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So, this is an ordinary differential equation.
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Whereas option C contains a partial derivative, and so it is known as a partial differential equation.
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You may see the abbreviations O.D.E and P.D.E used to describe ordinary and partial differential equations, respectively.
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So, our answer to the question, which of the following relationships is an ordinary differential equation, is B.
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A and D are not differential equations.
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And C is a differential equation, but it is a partial differential equation.