Video Transcript
Suppose that d𝑦 by d𝑥 is equal to
four 𝑥 minus four times the cos of two 𝑥 all divided by four times the sin of 𝑦
plus nine and 𝑦 is equal to zero when 𝑥 is equal to zero. Find 𝑦 in terms of 𝑥.
The question gives us a
differential equation, and it wants us to find an equation for 𝑦 in terms of 𝑥,
given that when 𝑦 is equal to zero, 𝑥 is equal to zero. We can see that d𝑦 by d𝑥 is the
product of a function in 𝑥 and a function in 𝑦. And we call first-order
differential equations, which have this property, separable. This is because we can separate the
𝑥-variables and the 𝑦-variables onto opposite sides of our equation. So to solve this, we’ll start by
multiplying both sides for our equation by four times the sin of 𝑦 plus nine. This gives us that four sin 𝑦 plus
nine multiplied by d𝑦 by d𝑥 is equal to four 𝑥 minus four times the cos of two
𝑥.
And at this point, it’s worth
reiterating, d𝑦 by d𝑥 is definitely not a fraction. However, when we’re solving
separable differential equations, we can treat it a little bit like a fraction. Doing this gives us the equivalent
statement four sin 𝑦 plus nine d𝑦 is equal to four 𝑥 minus four times the cos of
two 𝑥 d𝑥. Then we just take the integral of
both sides of this equation. And we see that these integrals are
in a form which we can solve. We know the integral of the sin of
𝜃 with respect to 𝜃 is equal to negative the cos of 𝜃 plus a constant of
integration 𝑐. So integrating four sin of 𝑦 gives
us negative four times the cos of 𝑦. And we know the integral of nine is
just nine 𝑦. Then we just add our constant of
integration we will call 𝑐 one.
Similarly, we know for constants 𝑎
and 𝑛, where 𝑛 is not equal to zero, the integral of 𝑎 times the cos of 𝑛𝜃 with
respect to 𝜃 is equal to 𝑎 times the sin of 𝑛𝜃 divided by 𝑛 plus the constant
of integration 𝑐. We integrate four 𝑥 by using the
power rule for integration. We add one to the exponent and
divide by this new exponent. This gives us four 𝑥 squared over
two, which is just two 𝑥 squared. We apply our integral rule to
integrate negative four times the cos of two 𝑥. This gives us negative four times
the sin of two 𝑥 divided by two, which is just negative two times the sin of two
𝑥.
And finally, we add a constant of
integration we will call 𝑐 two. We can simplify this. We can combine the constants 𝑐 one
and 𝑐 two into a new constant we will just call 𝑐. And we can find the value of 𝑐,
since we’re told that when 𝑦 is equal to zero, 𝑥 is equal to zero. Substituting 𝑥 is equal to zero
and 𝑦 is equal to zero gives us negative four times the cos of zero plus nine times
zero is equal to two times zero squared minus two times the sin of two times zero
plus 𝑐. By using the fact that the cos of
zero is equal to one and the sin of zero is equal to zero, we can simplify this
expression to see that 𝑐 is equal to negative four.
So by using 𝑐 is equal to negative
four, we must have that nine 𝑦 minus four times the cos of 𝑦 is equal to two 𝑥
squared minus two times the sin of two 𝑥 minus four. Therefore, we’ve shown if d𝑦 by
d𝑥 is equal to four 𝑥 minus four times the cos of two 𝑥 divided by four sin 𝑦
plus nine and 𝑦 is equal to zero when 𝑥 is equal to zero. Then we must have that nine 𝑦
minus four cos of 𝑦 is equal to two 𝑥 squared minus two times the sin of two 𝑥
minus four.
