https://file.aiquickdraw.com/mathbot/file/17269856857779gbrcp3s.jpg

Let's solve the given differential equation and analyze the behavior of the solution with the initial condition \( y(0) = 5 \).

### Given:
\[
\frac{dy}{dt} = (y - 5)(y + 3)
\]

This is a separable differential equation, which can be solved by analyzing its critical points and the behavior of \( y(t) \).

### Step 1: Critical points
The right-hand side of the equation is \( (y - 5)(y + 3) \). To find the equilibrium solutions, set this equal to zero:
\[
(y - 5)(y + 3) = 0
\]
This gives the equilibrium points:
\[
y = 5 \quad \text{and} \quad y = -3
\]

### Step 2: Behavior near critical points
- When \( y > 5 \), \( (y - 5)(y + 3) > 0 \), so \( \frac{dy}{dt} > 0 \), meaning \( y(t) \) is increasing.
- When \( -3 < y < 5 \), \( (y - 5)(y + 3) < 0 \), so \( \frac{dy}{dt} < 0 \), meaning \( y(t) \) is decreasing.
- When \( y < -3 \), \( (y - 5)(y + 3) > 0 \), so \( \frac{dy}{dt} > 0 \), meaning \( y(t) \) is increasing.

### Step 3: Initial condition \( y(0) = 5 \)
At \( y(0) = 5 \), the differential equation is:
\[
\frac{dy}{dt} = (5 - 5)(5 + 3) = 0
\]
This means that \( y = 5 \) is a constant solution.

### Conclusion:
The solution with the initial condition \( y(0) = 5 \) is constant because the derivative is zero at \( y = 5 \), meaning that the solution remains at \( y = 5 \) for all time.

The correct answer is:
**The solution with initial condition \( y(0) = 5 \) is constant.**

---

Do you need further clarification or have any other questions?

Here are 5 related questions to expand your understanding:
1. What is the general solution of the separable differential equation \( \frac{dy}{dt} = (y - 5)(y + 3) \)?
2. How do you classify equilibrium points in differential equations?
3. How does the phase line for this differential equation look like?
4. How does the solution behave for initial conditions \( y(0) > 5 \) or \( y(0) < -3 \)?
5. How would you find the solution to this equation analytically by integrating both sides?

**Tip:** Identifying the critical points of a differential equation helps you understand the long-term behavior of solutions and whether they are stable or unstable.

For the differential equation dy/dt = (y - 5)(y + 3), which of the following is true?

Explore the behavior of solutions to the differential equation dy/dt = (y - 5)(y + 3) based on different initial conditions. This problem involves identifying equilibrium solutions, analyzing the increasing and decreasing nature of the function, and determining the constant solutions for specific initial conditions. Suitable for college-level studies in mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation