Use separation of variables to solve dy dx − tan x = y2 tan x with y(0) = √3. Find the value of c in radians, not degrees

Answer:[tex]y(x)=tan(-log(cos(x))+\frac{\pi }{3} )[/tex]Step-by-step explanation:Rewrite the equation as:[tex]\frac{dy(x)}{dx}-tan(x)=y(x)^{2} *tan(x)[/tex]Isolating [tex]\frac{dy}{dx}[/tex][tex]\frac{dy}{dx} =tan(x)+tan(x)*y^{2}[/tex]Factor:[tex]\frac{dy}{dx} =tan(x)*(1+y^{2} )[/tex]Dividing both sides by [tex](1+y^{2} )[/tex] and multiplying them by [tex]dx[/tex][tex]\frac{dy}{1+y^{2} } =tan(x)dx[/tex]Integrate both sides:[tex]\int\ \frac{dy}{1+y^{2} } = \int\ tan(x)  dx[/tex]Evaluate the integrals:[tex]arctan(y)=-log(cos(x))+C_1[/tex]Solving for y:[tex]y(x)=tan(-log(cos(x))+C_1)[/tex]Evaluating the initial condition:[tex]y(0)=\sqrt{3} =tan(-log(cos(0))+C_1)=tan(-log(1)+C_1)=tan(0+C_1)[/tex][tex]\sqrt{3} =tan(C_1)\\arctan(\sqrt{3} )=C_1\\60=C_1[/tex]Converting 60 degrees to radians:[tex]60degrees*\frac{\pi }{180degrees} =\frac{\pi }{3}[/tex]Replacing [tex]C_1[/tex] in the diferential equation solution:[tex]y(x)=tan(-log(cos(x))+\frac{\pi }{3} )[/tex]