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
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.