Question

Use separation of variables to find the solution to the differential equation subject to the given initial condition.

dP/dt = 9P, P(0) = 6

Answer 1

To solve the given differential equation using separation of variables, separate the variables and integrate both sides. Apply the initial condition P(0) = 6 to find the constant of integration and express P as a function of t, resulting in P(t) = 6e^9t.

Step-by-step explanation:

To solve the differential equation dP/dt = 9P with the initial condition P(0) = 6 using separation of variables, follow these steps:

Rearrange the equation to separate the variables, placing all terms involving P on one side and all terms involving t on the other.

Integrate both sides of the equation.

Apply the initial condition to solve for the constant of integration.

Write down the solution for P as a function of time t.

Here is the step-by-step process:

1. Separate variables: dP/P = 9 dt.

2. Integrate both sides: ln(P) = 9t + C.

3. Solve for P using the initial condition:

4. When t = 0, P = 6.

5. Therefore, ln(6) = C.

6. Exponentiate both sides to solve for P: P(t) = e(9t + ln(6)), which simplifies to P(t) = 6e9t.

The solution to the differential equation with the given initial condition is P(t) = 6e9t.