Video Transcript
Which of the following is the
solution to the differential equation d𝑦 by d𝑥 equals five cos 𝑥 with the initial
condition 𝑦 at 𝜋 by two is equal to four? Is it a) 𝑦 equals five sin 𝑥
minus four? Is it b) 𝑦 equals negative five
sin 𝑥 plus nine? Is it c) 𝑦 equals five sin 𝑥
minus one, d) 𝑦 equals negative five sin 𝑥 minus nine, or e) 𝑦 equals five sin 𝑥
plus one?
A differential equation is an
equation which contains derivatives. Our job then is to solve the
differential equation d𝑦 by d𝑥 equals five cos 𝑥. And this is going to involve
integration at some point. Here I have d𝑦 by d𝑥 as some
function of 𝑥. In this case, we can simply
integrate both sides of our equation with respect to 𝑥. The integral of the derivative of
𝑦 with respect to 𝑥 with respect to 𝑥 is 𝑦.
On the right-hand side, we can take
the five outside of the integral sign and focus on integrating cos 𝑥 with respect
to 𝑥. We know that the integral of cos 𝑥
with respect to 𝑥 is sin 𝑥 plus that constant of integration. So we can see then that 𝑦 is equal
to five sin 𝑥 plus 𝑐, that constant of integration.
But we have the condition 𝑦 of 𝜋
by two is equal to four. In other words, when 𝑥 is equal to
𝜋 by two, 𝑦 is four. So we can substitute these values
into our solution to the differential equation to find the value of 𝑐. That gives us four equals five sin
of 𝜋 by two plus 𝑐. And of course sin of 𝜋 by two is
one.
So we have four equals five times
one plus 𝑐. Well, five times one is just
five. And we can solve for 𝑐 by
subtracting five from both sides of this equation. And we see that 𝑐 is equal to
negative one. We replace 𝑐 with negative one in
the equation 𝑦 equals five sin 𝑥 plus 𝑐. And we see that the solution to the
differential equation d𝑦 by d𝑥 equals five cos 𝑥 with that initial condition is
𝑐. It’s 𝑦 equals five sin 𝑥 minus
one.
