# Thread: A domain restriction on the solution of a differential equation

1. ## A domain restriction on the solution of a differential equation

I'm an AP Calc teacher. A problem I gave my students states:

Find the general solution to the exact differential equation.

$\displaystyle y'=5 sec^2 x - 1.5 \sqrt x$

$\displaystyle y(0)=7$

So taking the anti derivative,

$\displaystyle y=5 tan x - x^{3/2} + 7$

That's fine. The solution also notes:

$\displaystyle (0 < x < \pi/2)$

Can anyone explain why this domain restriction is in place? I understand that there is a discontinuity at pi/2, but why does it choose this specific interval? Is it because the initial value is given at 0?

Thanks for any help.

2. ## Re: A domain restriction on the solution of a differential equation

at pi/2 you would be dividing by 0 in the differential equation.

I'd think, since an initial condition is given at 0, that the actual domain is [0, pi/2)

You could choose any domain that includes zero and doesn't include (k+1/2)pi, k an integer, and those are
(-pi/2, 0], and [0, pi/2), however you also have a square root of x in your diff eq. That rules out the first interval if you are working on the real numbers.

So that leaves you only with [0,pi/2)