# What is the particular solution of the differential equation  dy/dt = -10t+1  with Initial Condition y(0)=-5?

May 4, 2017

$y = - 5 {t}^{2} + t - 5$

#### Explanation:

We have:

$\frac{\mathrm{dy}}{\mathrm{dt}} = - 10 t + 1$

This is a First Order Separable Differential Equation in standard form, so we can "separate the variables" to get:

$\int \setminus \mathrm{dy} = \int \setminus - 10 t + 1 \setminus \mathrm{dt}$

Which we can integrate to get:

$y = - 5 {t}^{2} + t + C$

And using the Initial Condition $y \left(0\right) = - 5$ we have:

$- 5 = 0 + 0 + C \implies C = - 5$

Hence the unique solution is:

$y = - 5 {t}^{2} + t - 5$