# Integration by separation of variables: algebraic rearrangement?

## A lake contains 5,000,000 million litres of unpolluted water. A river flows into the lake at 100,000 litres per day. Due to polluters, the river flowing in contains 5 grams per litre of pollutant. A river flows out of the lake at 100,000 litres per day. Find an expression for the amount of pollutant in the lake. I have: $\frac{\mathrm{dp}}{\mathrm{dt}} = 500 , 000 - \frac{p}{50}$ $p = 25 , 000 , 000 + {e}^{- \frac{t}{50} + c}$ Answer says $p = 25 , 000 , 000 \left(1 - {e}^{- \frac{t}{50}}\right)$

May 13, 2016

your answer is almost correct. Needs to get rid of c only, as explained below.

#### Explanation:

Your derivation is ok. Only thing left is to determine the constant of integration c.

For this apply the initial condition that at t=0, p=0 (there was no pollution initially)

Thus $0 = 25 , 000 , 000 + {e}^{c}$

Thus ${e}^{c} = - 25 , 000 , 000$. Your answer would then become

$p = 25 , 000 , 000 + {e}^{- \frac{t}{50}} . {e}^{c}$

$p = 25 , 000 , 000 - 25 , 000 , 000 {e}^{- \frac{t}{50}}$

$= 25 , 000 , 000 \left(1 - {e}^{- \frac{t}{50}}\right)$