### Video Transcript

Determine the order of the differential equation 𝑦 to the 10th power minus four d𝑥 by d𝑦 plus d two 𝑥 by d𝑦 squared to the seventh power is equal to the sin of 𝑥.

We’re given a differential equation, and we’re asked to determine the order of this differential equation. So let’s start by recalling what we mean by the order of a differential equation. The order of a differential equation is the order of the highest-order derivative in the equation. So to find the order of our differential equation, we first need to find the highest-order derivative which appears in this equation.

We can just look at each term individually. First, let’s look at 𝑦 to the 10th power. We can see this doesn’t contain any derivatives. In our second term, we can see we have one derivative expression, d𝑥 by d𝑦.

Now, there’s a few things we might be worried about here. For example, we’re used to seeing d𝑥 by d𝑦, the derivative of 𝑦 with respect to 𝑥. However, this being d𝑥 by d𝑦 doesn’t change anything. We’re just taking the derivative of 𝑥 with respect to 𝑦. And the most important thing to notice here is this is the first derivative of 𝑥 with respect to 𝑦. This means d𝑥 by d𝑦 is a first-order derivative.

We can now move on to our next term, d two 𝑥 by d𝑦 squared all raised to the seventh power. Once again, we notice that d two 𝑥 by d𝑦 squared is a differential. However, this time, we need to recall this is the second derivative of 𝑥 with respect to 𝑦. And this means this term is of the second order. We might be worried that we’re raising this to the seventh power. However, we need to remember that raising this to the seventh power does not mean we’re taking any more derivatives of 𝑥. So this won’t change the order. Finally, our last term is the sin of 𝑥. This doesn’t include any derivatives. So it won’t change the order of our differential equation.

Now, remember, we need to find the order of the highest-order derivative in this equation. We see that this is the second derivative of 𝑥 with respect to 𝑦. So this means we’ve shown the order of this differential equation is two, and this is our final answer.

Therefore, by finding the highest-order derivative in the differential equation, we were able to show the order of the differential equation given to us in the question is two.