Question Video: Deciding Which of a List of Differential Equations Is of the Second Order Mathematics • Higher Education

Video Transcript

Which of the following differential equations is of the second order? Option (A) four 𝑦 cubed minus nine 𝑥 times d two 𝑦 by d𝑥 squared is equal to one plus 𝑒 to the power of three 𝑥. Option (B) four 𝑦 squared minus nine 𝑥 times d𝑦 by d𝑥 is equal to one plus 𝑒 to the power of three 𝑥. Option (C) four 𝑦 cubed minus nine 𝑥 times d three 𝑦 by d𝑥 cubed is equal to one plus 𝑒 to the power of two 𝑥. Option (D) four 𝑦 cubed minus nine 𝑥 squared times d𝑦 by d𝑥 is equal to one plus 𝑒 to the power of three 𝑥. Or option (E) four 𝑦 cubed minus two times d𝑦 by d𝑥 is equal to one plus 𝑒 to the power of three 𝑥.

We’re given five differential equations, and we need to determine which of these five is of the second order. To answer this question, let’s start by recalling what we mean by the order of a differential equation.

We recall the order of a differential equation is the order of the highest-order derivative which appears in the equation. In other words, to determine the order of a differential equation, we just need to find the highest-order derivative which appears in this equation. So one way of approaching this problem is just to find the order of all five of the differential equations given to us in the options.

Let’s start with option (A). We can do this term by term. The first term is four 𝑦 cubed. Four 𝑦 cubed does not contain any derivatives. So the order of this term alone is just equal to zero. Our next term is nine 𝑥 times d two 𝑦 by d𝑥 squared. We know nine 𝑥 does not contain any derivatives. However, d two 𝑦 by d𝑥 squared is the second derivative of 𝑦 with respect to 𝑥. And because this is the second derivative of 𝑦 with respect to 𝑥, we say that its order is two.

We do still need to check the rest of the terms in this expression. If we look at our last two terms of one and 𝑒 to the power of three 𝑥, we can see neither of these terms contain a derivative. And this tells us we found the highest-order differential in this expression. It was d two 𝑦 by d𝑥 squared which had order two. And if the highest-order differential which appears in this equation is two, that means the order of this differential equation must be two. Therefore, we’ve shown the differential equation in option (A) is of the second order.

We could leave our answer like this. However, we can also check the order of the other five options given to us in the question. In option (B), we see the terms four 𝑦 squared, one, and 𝑒 to the power of three 𝑥 do not contain any differentials. However, we see our second term contains d𝑦 by d𝑥. That’s the first derivative of 𝑦 with respect to 𝑥. This means the highest-order derivative which appears in this equation is a first derivative. So we say that the equation in option (B) is order one.

We get a similar story in option (C). We can see the terms four 𝑦 cubed, one, and 𝑒 to the power of two 𝑥 do not contain any derivatives. However, our second term contains d three 𝑦 by d𝑥 cubed. That’s the third derivative of 𝑦 with respect to 𝑥. So the highest-order derivative which appears in this expression is a third derivative. So we say it’s order three or of the third order.

And if we look at option (D) and option (E), we get exactly the same story as in option (B). We see that four 𝑦 cubed, one, and 𝑒 to the power of three 𝑥 do not contain any derivatives. However, both of our second terms contain one derivative. We get d𝑦 by d𝑥. And we know d𝑦 by d𝑥 is the first derivative of 𝑦 with respect to 𝑥. So both option (D) and option (E) contain a highest-order derivative of one.

Therefore, we were able to show that only option (A) is a second-order differential equation.