# What is a particular solution to the differential equation dy/dx=x/y with y(1)=2?

Jul 31, 2016

$y = \sqrt{{x}^{2} + 3}$

#### Explanation:

this is separable

$y ' = \frac{x}{y}$

$y \setminus y ' = x$

$\int \setminus y \setminus y ' \setminus \mathrm{dx} = \int \setminus x \setminus \mathrm{dx}$

$\int \setminus \frac{d}{\mathrm{dx}} \left({y}^{2} / 2\right) \setminus \mathrm{dx} = \int \setminus x \setminus \mathrm{dx}$

${y}^{2} / 2 = {x}^{2} / 2 + C$

${y}^{2} = {x}^{2} + C$

$y = \pm \sqrt{{x}^{2} + C}$

applying IV: $y \left(1\right) = 2$

$2 = \pm \sqrt{1 + C}$

suggest the + root solution. so

$2 = \sqrt{1 + C} \implies C = 3$

$y = \sqrt{{x}^{2} + 3}$