Hi its me again and I have return with more question that I currently got stuck on

************************************************** *********

For the following differential equation

$\displaystyle \frac{dy}{dx}=\frac{2cos^2(x)-sin^2(x)+y^2}{2cos(x)}$ , $\displaystyle \frac{-\pi}{2}<x<\frac{\pi}{2}$

show that the substitution $\displaystyle y(x)=sin(x)+\frac{1}{u(x)}$ , yields the differential equation for $\displaystyle u(x)$

$\displaystyle \frac{du}{dx}=-u*tan(x)-\frac{1}{2}sec(x)$

Hence find the solution $\displaystyle y(x)$ to the original differential equation that satisfies the condition $\displaystyle y(0)=2$. Find the interval on which the solution to the initial value problem is defined.

************************************************** **********

So I first set out to integrate $\displaystyle \frac{dy}{dx}$ first, however I have a problem that I do not know how to move $\displaystyle y^2$ to the otherside of the equation

e.g. $\displaystyle \frac{1}{y^2}dy=\frac{2cos^2(x)-sin^2(x)}{2cos(x)}dx$

what I did is

$\displaystyle \int 1 dy = \int cos(x)dx -\frac{1}{2}\int sin(x)tan(x)dx + \frac{y^2}{2}\int\frac{1}{cos(x)}dx$

I integrated $\displaystyle \int sin(x)tan(x)dx$ by using integration by parts with u=tan(x) and $\displaystyle \frac{dv}{dx}=sin(x)$

so the equation become

$\displaystyle y=sin(x)+\frac{cos(x)tan(x)}{2}-\frac{sin(x)}{2}+\frac{y^2}{2sin(x)}$ + C

I'm certain that my integration is wrong, since y(0)=2 will result in $\displaystyle \frac{y^2}{0}$

The same problem also applies to $\displaystyle \frac{du}{dx}=-u*tan(x)-\frac{1}{2}sec(x)$

since I do not know how to move, the $\displaystyle y$ and $\displaystyle u$ to the other side of the equation. So can any body tell me how to integrate this question?

Thanks

Junks

P.S. I finally posted this in latex

edit: I'll try to use First Order, O.D.E but since $\displaystyle y^2$will first order O,D,E work there?

By first order O.D. E

$\displaystyle \frac{du}{dx}=-u*tan(x)-\frac{1}{2}sec(x)$

I found my $\displaystyle u$ to be$\displaystyle u=\frac{-1}{2}cos(x)tan(x)$

is this correct?