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.
