Is the function 𝑦 equals one over
two plus 𝑥 a solution to the differential equation 𝑦 prime equals negative 𝑦
Remember that 𝑦 prime is another
way of saying d𝑦 by d𝑥, the first derivative of 𝑦 with respect to 𝑥. So, we’ve been given a first order
differential equation, and we want to know whether the given function 𝑦 is a
solution to it. That is, we need to know whether
the function 𝑦 satisfies this equation.
Let’s begin then by first working
out what 𝑦 prime, or d𝑦 by d𝑥, is equal to for this function 𝑦. And to do this, we can first
express 𝑦 in an alternative form. We can write it as two plus 𝑥 to
the power of negative one. We can then find this derivative
using the general power rule, which says that if we have some function 𝑓 of 𝑥 to
the power of 𝑛, then its derivative with respect to 𝑥 is equal to 𝑛 multiplied by
𝑓 prime of 𝑥 multiplied by 𝑓 of 𝑥 to the power of 𝑛 minus one.
Here, our function 𝑓 of 𝑥 is two
plus 𝑥, and our power 𝑛 is negative one. So, applying the general power
rule, we have 𝑛, that’s negative one, multiplied by the derivative of two plus 𝑥,
which is just one, multiplied by 𝑓 of 𝑥. That’s two plus 𝑥, to the power of
𝑛 minus one. So, that’s the power of negative
two. We can then rewrite this as
negative one over two plus 𝑥 all squared. So, we know what the left-hand side
of this differential equation would be for this function 𝑦.
On the right-hand side, we have
negative 𝑦 squared. So, that’s the original function 𝑦
squared and then multiply it by negative one, which is equal to negative one over
two plus 𝑥 all squared. To square a fraction, we can square
the numerator and square the denominator. So, we have negative one squared,
which is one, over two plus 𝑥 all squared.
Now, we compare our expressions for
𝑦 prime and negative 𝑦 squared. And we see that they are both equal
to negative one over two plus 𝑥 all squared. And therefore, they are indeed
equal to one another. This tells us that the function 𝑦
equals one over two plus 𝑥 does satisfy the given differential equation and,
therefore, it is a solution.