Question

Find the particular solution to y ' = 4sin(x) given the general solution is y = c - 4cos(x) and the initial condition y of pi over 2 equals 2

Answer 1

The general solutions always have some additive/multiplicative constant, that you must fix in the particular solution.

In order to do so, you need to impose that the particular solution passes through a certain point. In your case, you have

`y(x) = c-4\cos(x)`

and you want

`y\left((\pi)/(2)\right) = 2`

Put everything together, and you have

`y\left((\pi)/(2)\right) = c-4\cos\left((\pi)/(2)\right) = c = 2`

Since the cosine is zero in the chosen point. So, we've fixed the value of the constant, and the particular solution is found:

`y(x) = 2-4\cos(x)`