What is a solution to the differential equation dy/dx=(1+x)/(xy) with y(1)=-4?

Jul 11, 2016

$\implies y = - \sqrt{2 \ln x + 2 x + 14}$

Explanation:

separable equation so separate

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1 + x}{x y}$

$y \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1 + x}{x}$

$y \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{x} + 1$

$\int \setminus y \setminus \frac{\mathrm{dy}}{\mathrm{dx}} \setminus \mathrm{dx} = \int \setminus \frac{1}{x} + 1 \setminus \mathrm{dx}$

$\int \setminus y \setminus \mathrm{dy} = \int \setminus \frac{1}{x} + 1 \setminus \mathrm{dx}$

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

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

$y \left(1\right) = - 4$

$- 4 = \pm \sqrt{0 + 2 + C}$

$\implies y = - \sqrt{2 \ln x + 2 x + 14}$