# What is a solution to the differential equation dy/dx=sqrt(xy)sinx?

Jan 8, 2017

$y = {\left(\frac{1}{2} \int {x}^{\frac{1}{2}} \sin x \mathrm{dx}\right)}^{2}$

#### Explanation:

We restrict ourselves to $x > 0$, $y > 0$ so that:

$\sqrt{x y} = \sqrt{x} \sqrt{y}$ and the variables are separable:

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

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

$\int {y}^{- \frac{1}{2}} \mathrm{dy} = \int {x}^{\frac{1}{2}} \sin x \mathrm{dx}$

$2 \sqrt{y} = \int {x}^{\frac{1}{2}} \sin x \mathrm{dx}$

The right hand integral can be reduced to a Fresnel integral and cannot be expressed through elementary functions.