Find the general integral of the Bernoulli equation…

Find the general integral of the Bernoulli equation $y'+2\dfrac{y}{x}=\dfrac{2\sqrt{y}}{\cos^2x}$.

Explanation of Method

With $n=\tfrac12$, substitute $v=\sqrt{y}$ (so $y=v^2$, $y'=2vv'$) to reduce to a linear equation for $v(x)$.

Step 1: Substitute $v=\sqrt{y}$

With $y=v^2$, $y'=2vv'$: $2vv'+\dfrac{2v^2}{x}=2v\sec^2x$. Dividing by $2v$: $v'+\dfrac{v}{x}=\sec^2x$.

Step 2: Solve the linear equation

With $\mu(x)=x$: $\dfrac{d}{dx}(xv)=x\sec^2x$. By parts, $\displaystyle\int x\sec^2x\,dx=x\tan x+\ln|\cos x|+C$, so $v=\tan x+\dfrac{\ln|\cos x|+C}{x}$.

Result

Final answer:

\[\boxed{\,\sqrt{y}=\tan x+\frac{\ln|\cos x|+C}{x},\qquad C\in\mathbb{R}}\]