Answer
The solution to the differential equation is \( P(t) = \frac{4200 e^{3.6 t}}{293 + 7 e^{3.6 t}} \). To verify, plot this function and compare it with the slope field generated by a graphing utility; the solution should align with the direction indicated by the slope field, confirming its validity.
Solution
To solve the initial value problem:
\[
\frac{dP}{dt} = 0.006 P (600 - P), \quad P(0) = 14
\]
we will use the method of **separation of variables** along with **partial fractions**. After finding the explicit solution, we'll discuss how to verify it using a slope field generated by a graphing utility.
### Step 1: Separate Variables
Start by separating the variables \( P \) and \( t \):
\[
\frac{dP}{P(600 - P)} = 0.006 \, dt
\]
### Step 2: Apply Partial Fractions
To integrate the left-hand side, decompose the fraction using partial fractions:
\[
\frac{1}{P(600 - P)} = \frac{A}{P} + \frac{B}{600 - P}
\]
Multiply both sides by \( P(600 - P) \):
\[
1 = A(600 - P) + B P
\]
To find constants \( A \) and \( B \):
1. Let \( P = 0 \):
\[
1 = A(600) \implies A = \frac{1}{600}
\]
2. Let \( P = 600 \):
\[
1 = B(600) \implies B = \frac{1}{600}
\]
Thus:
\[
\frac{1}{P(600 - P)} = \frac{1}{600} \left( \frac{1}{P} + \frac{1}{600 - P} \right)
\]
### Step 3: Integrate Both Sides
Integrate the separated equation:
\[
\int \frac{1}{P(600 - P)} \, dP = \int 0.006 \, dt
\]
Substitute the partial fractions:
\[
\frac{1}{600} \int \left( \frac{1}{P} + \frac{1}{600 - P} \right) dP = 0.006 \int dt
\]
Integrate:
\[
\frac{1}{600} \left( \ln|P| - \ln|600 - P| \right) = 0.006 t + C
\]
Multiply both sides by 600:
\[
\ln\left(\frac{P}{600 - P}\right) = 3.6 t + C'
\]
where \( C' = 600C \) is a new constant.
### Step 4: Solve for \( P(t) \)
Exponentiate both sides to eliminate the natural logarithm:
\[
\frac{P}{600 - P} = Ce^{3.6 t}
\]
where \( C = e^{C'} \).
Solve for \( P \):
\[
P = \frac{600 C e^{3.6 t}}{1 + C e^{3.6 t}}
\]
### Step 5: Apply Initial Condition
Use the initial condition \( P(0) = 14 \) to find \( C \):
\[
14 = \frac{600 C}{1 + C} \implies 14(1 + C) = 600C \implies 14 + 14C = 600C
\]
\[
14 = 586C \implies C = \frac{14}{586} = \frac{7}{293}
\]
### Step 6: Write the Explicit Solution
Substitute \( C = \frac{7}{293} \) back into the equation for \( P(t) \):
\[
P(t) = \frac{600 \cdot \frac{7}{293} e^{3.6 t}}{1 + \frac{7}{293} e^{3.6 t}} = \frac{4200 e^{3.6 t}}{293 + 7 e^{3.6 t}}
\]
Thus, the explicit solution is:
\[
P(t) = \frac{4200 e^{3.6 t}}{293 + 7 e^{3.6 t}}
\]
Alternatively, it can be expressed as:
\[
P(t) = \frac{600}{1 + \left(\frac{293}{7}\right) e^{-3.6 t}}
\]
### Step 7: Verifying with a Slope Field
To verify that the solution conforms to the slope field, follow these steps using a graphing utility (e.g., Desmos, GeoGebra, or a graphing calculator):
1. **Generate the Slope Field:**
- Input the differential equation \( \frac{dP}{dt} = 0.006 P (600 - P) \) into the slope field generator.
- The slope field visually represents the direction of \( dP/dt \) at various points \( (t, P) \).
2. **Plot the Solution Curve:**
- Use the explicit solution \( P(t) = \frac{4200 e^{3.6 t}}{293 + 7 e^{3.6 t}} \) to plot \( P \) versus \( t \).
- Alternatively, use the logistic function form \( P(t) = \frac{600}{1 + \left(\frac{293}{7}\right) e^{-3.6 t}} \).
3. **Verification:**
- Observe that the solution curve aligns smoothly with the direction indicated by the slope field at all points.
- The initial condition \( P(0) = 14 \) should lie on both the solution curve and the slope field.
- As \( t \) increases, \( P(t) \) approaches the carrying capacity \( 600 \), which should coincide with horizontal slopes in the slope field as \( P \) nears \( 600 \).
### Example Using Desmos
Here's how you can perform the verification using Desmos:
1. **Access Desmos:**
- Go to [Desmos Graphing Calculator](https://www.desmos.com/calculator).
2. **Input the Slope Field:**
- Unfortunately, Desmos doesn't have a built-in slope field feature, but you can approximate it by plotting many small line segments with slopes given by the differential equation.
- Alternatively, use a dedicated slope field tool or software like GeoGebra.
3. **Plot the Solution:**
- Enter the explicit solution formula:
\[
P(t) = \frac{4200 e^{3.6 t}}{293 + 7 e^{3.6 t}}
\]
- In Desmos, type:
```
P(t) = 4200 * e^(3.6t) / (293 + 7 * e^(3.6t))
```
- Plot this function to see the logistic growth curve.
4. **Compare with the Slope Field:**
- If using separate software for the slope field, overlay or compare the solution curve with the slope field.
- The curve should follow the directions indicated by the slope field, confirming that it is indeed a solution to the differential equation.
### Conclusion
The solution to the initial value problem is:
\[
P(t) = \frac{4200 e^{3.6 t}}{293 + 7 e^{3.6 t}} \quad \text{or} \quad P(t) = \frac{600}{1 + \left(\frac{293}{7}\right) e^{-3.6 t}}
\]
By generating and comparing this solution with the slope field using a graphing utility, you can visually verify that the solution conforms to the behavior dictated by the differential equation.
Reviewed and approved by the UpStudy tutoring team
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