13. Intro to Differential Equations

Separate the variables of the following differential equation.

$ysi{n}^2\theta \frac{dy}{d\theta }=1$

Find the particular solution that satisfies the given initial condition

$(y^2+y)e^x{y}^{\prime }=y^3+e^x{y}^3;y\left(0\right)=1$

Find the general solution to the differential equation.

$\frac{dy}{dx}=y\sqrt{x}$

Find the particular solution that satisfies the given initial condition $\frac{dy}{dx}=\sin x\bullet \sec y,y\left(\frac{\pi }{2}\right)=\frac{\pi }{4}$.