# Video: AP Calculus AB Exam 1 • Section I • Part A • Question 29

Which of the following is the solution to the differential equation d𝑦/d𝑥 = 5 cos 𝑥 with the initial condition 𝑦(𝜋/2) = 4? [A] 𝑦 = 5 sin 𝑥 − 4 [B] 𝑦 = −5 sin 𝑥 + 9 [C] 𝑦 = 5 sin 𝑥 − 1 [D] 𝑦 = −5 sin 𝑥 − 9 [E] 𝑦 = 5 sin 𝑥 + 1

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### 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.