# What is a solution to the differential equation dx/dt=t(x-2)?

Jul 11, 2016

$x = C {e}^{{t}^{2} / 2} + 2$

#### Explanation:

this is separable

$\frac{\mathrm{dx}}{\mathrm{dt}} = t \left(x - 2\right)$

$\frac{1}{x - 2} \setminus \frac{\mathrm{dx}}{\mathrm{dt}} = t$

$\int \setminus \frac{1}{x - 2} \setminus \frac{\mathrm{dx}}{\mathrm{dt}} \setminus \mathrm{dt} = \int \setminus t \setminus \mathrm{dt}$

$\int \setminus \frac{1}{x - 2} \setminus \mathrm{dx} = \int \setminus t \setminus \mathrm{dt}$

$\ln \left(x - 2\right) = {t}^{2} / 2 + C$

$x - 2 = {e}^{{t}^{2} / 2 + C} = C {e}^{{t}^{2} / 2}$

$x = C {e}^{{t}^{2} / 2} + 2$