# How to solve the seperable differential equation and when using the following initial condition: y(1)=2 ?

May 9, 2016

$y = \sqrt[6]{6 \ln \left(x\right) + 6 x + 58}$

#### Explanation:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1 + x}{x {y}^{5}}$

$\implies {y}^{5} \mathrm{dy} = \frac{1 + x}{x} \mathrm{dx} = \left(\frac{1}{x} + 1\right) \mathrm{dx}$

$\implies \int {y}^{5} \mathrm{dy} = \int \left(\frac{1}{x} + 1\right) \mathrm{dx}$

$\implies {y}^{6} / 6 = \ln \left(x\right) + x + C$

$\implies {y}^{6} = 6 \ln \left(x\right) + 6 x + C$

By the initial condition of $y \left(1\right) = 2$, we have

${2}^{6} = 6 \ln \left(1\right) + 6 + C = 6 + C$

$\implies C = {2}^{6} - 6 = 58$

$\implies {y}^{6} = 6 \ln \left(x\right) + 6 x + 58$

$\therefore y = \sqrt[6]{6 \ln \left(x\right) + 6 x + 58}$