Solution verification on homework problem. Separable first order ODE IVP. The answer is supposedly $y^2 = 1 + \sqrt{x^2 - 16}$ I don't know where I went wrong cause I know for a fact that my substitution of $x = 4 \sec(\theta)$ is correct. I know for a fact that after substitution the integral becomes $\int \sec^2(\theta)= \tan(\theta)+C$. I am pretty sure that my substitution of $\tan(\theta)$ is correct. Am I missing something?
$2y \frac{dy}{dx} = \frac{x}{\sqrt{x^2 - 16}}  \space \space \space y(5)=2$
$\int 2y \space dy = \int \frac{4\sec(\theta)}{\sqrt{(4\sec(\theta))^2-16}}\space d(\theta)$
$y^2 = \int \frac{4\sec(\theta)}{\sqrt{(\sec^2(\theta)-1)16}}\sec(\theta) \ tan(\theta)$
$y^2 = \frac{4\sec(\theta)}{4\tan(\theta)}\sec(\theta)\tan(\theta)$
$y^2 = \int sec^2(\theta)$
$y^2 = \tan(\theta) + c$
Using the reference triangle: Tangent is equal to $\frac{\sqrt{x^2-16}}{4}$
$y^2= \frac{\sqrt{x^2-16}}{4} + C$
$4 = \frac{3}{4}+ C$
$C = \frac{15}{4}$
$y^2= \frac{\sqrt{x^2-16}}{4} + \frac{15}{4}$

In integrating the RHS of  in$$2y \frac{dy}{dx} = \frac{x}{\sqrt{x^2 - 16}}  \space \space \space y(5)=2$$
it is advised to let $$u=x^2-16$$ and you have $$du=2xdx$$
Then you do not have to worry about trig substitution at all. 
