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]
