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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 $\frac{\sqrt{x^2-16}}{4}$

$y^2= \frac{\sqrt{x^2-16}}{4} + C$

$4 = \frac{3}{4}+ C$

$C = \frac{15}{4}$

$y^2= \frac{\sqrt{x^2-16}}{4} + \frac{15}{4}$

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You have to use another $4$ .
$x = 4\sec\theta \implies dx = \color{red}4\sec\theta\tan\theta d\theta$

So, you'll have $y^2 = \sqrt{x^2-16} +c$

$2^2 = \sqrt{9} +c \implies c = 4-3= 1$

Thus,

$$y^2 =\sqrt{x^2+16} +1$$

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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.

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