# What is a solution to the differential equation dy/dx=2y-1?

Aug 23, 2016

$y = \frac{{C}_{2} {e}^{2 x} + 1}{2}$

#### Explanation:

Substituting $z = 2 y - 1$ in $\frac{\mathrm{dy}}{\mathrm{dx}} = 2 y - 1$

and considering

$\frac{\mathrm{dz}}{\mathrm{dx}} = 2 \frac{\mathrm{dy}}{\mathrm{dx}}$ we have

$\frac{1}{2} \frac{\mathrm{dz}}{\mathrm{dx}} = z$. Grouping variables

$\frac{\mathrm{dz}}{z} = 2 \mathrm{dx}$ and integrating

${\log}_{e} z = 2 x + {C}_{1}$ or equivalently

$z = {C}_{2} {e}^{2 x} = 2 y - 1$ and finally

$y = \frac{{C}_{2} {e}^{2 x} + 1}{2}$