Use separation of variables to find the solution to the differential equation $\frac{d P}{d t}=0.06 P$ if $P(0)=90$.
\[
P(t)=
\]
Answer
The solution to the differential equation \(\frac{d P}{d t}=0.06 P\) with initial condition \(P(0)=90\) is \(P(t) = 90 * e^{0.06t}\). For example, at time \(t=10\), the population is approximately \(\boxed{163.99}\).
Step 1 ：This is a simple first order differential equation. The general solution to such an equation is given by the formula \(P(t) = P(0) * e^{kt}\), where \(P(0)\) is the initial condition, \(k\) is the rate of growth, and \(t\) is time. In this case, \(P(0) = 90\) and \(k = 0.06\). We can substitute these values into the formula to find the solution.
Step 2 ：Now that we have the function defined, we can use it to find the population at any given time. For example, if we want to find the population at time \(t = 10\), we can simply call the function \(P(10)\).
Step 3 ：The solution to the differential equation \(\frac{d P}{d t}=0.06 P\) with initial condition \(P(0)=90\) is \(P(t) = 90 * e^{0.06t}\). For example, at time \(t=10\), the population is approximately \(\boxed{163.99}\).
