Integration by separation of variables: algebraic rearrangement?

Question

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:
#(dp)/dt = 500,000-p/50#
#p=25,000,000+e^(-t/50+c)#

Answer says #p=25,000,000(1-e^(-t/50))#

Elijah Abram · Accepted Answer

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

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^(-t/50) . e^c# #p= 25,000,000-25,000,000 e^(-t/50)# #=25,000,000(1-e^(-t/50))#

Aurora Alvarado · Answer

undefined Integration by separation of variables is a technique used to solve certain types of first-order ordinary differential equations. The process involves isolating variables on one side of the equation and integrating with respect to each variable separately. This typically requires algebraic rearrangement of terms to group variables together and constants on one side of the equation. Once the variables are separated, integration can be performed to find the solution. The resulting solution often involves an implicit equation or may require further manipulation to solve for a specific variable explicitly. Overall, algebraic rearrangement is a crucial step in the integration by separation of variables method.