# Lumyn Electronics has seen stellar expansion in recent months. The company's profit, P, is...

## Question:

Lumyn Electronics has seen a stellar expansion in recent months. The company's profit, P, is currently growing exponentially at 5% each year. The rate of change of the company's profit can be modeled as (1/P) dP=0.05dt.

Find a general formula that expresses the profit of the company as a function of time, t.

## Separable Differential Equations:

A differential equation is one that contains a derivative, i.e. one that involves differentials. For differential equations like the one above, variables are already separated so that only one appears on either side of the equation. To solve it, we integrate each side and simplify.

We integrate each side of the differential equation to find

{eq}\begin{align*} \int \frac1P \ dP &= \int 0.05 \ dt \\ \ln P &= 0.05t + C \\ P &= A e^{0.05t} \end{align*} {/eq}

where we wrote {eq}A = e^C {/eq}. Now, note that at time {eq}t = 0 {/eq}, we have {eq}P = A {/eq}, so {eq}A {/eq} is the starting profit, which we will call {eq}P_0 {/eq}. Then the profit function is

{eq}\begin{align*} P (t) &= P_0 e^{0.05t} \end{align*} {/eq}