Let's analyze the questions and solutions step-by-step:

Question 5: Solve the differential equation

The given differential equation is:

$\frac{dy}{dx} = \frac{3y}{x}$

with the initial condition $y(1) = -1$.

Step 1: Solve the differential equation

This is a separable differential equation, so separate variables:

$\frac{dy}{y} = \frac{3}{x} \, dx$

Integrate both sides:

$\int \frac{1}{y} \, dy = \int \frac{3}{x} \, dx$

This gives:

$\ln |y| = 3 \ln |x| + C$

Simplify:

$\ln |y| = \ln |x^3| + C$

Exponentiate both sides to remove the logarithm:

$|y| = e^C \cdot |x^3|$

Let $e^C = k$, where $k$ is an arbitrary constant. Thus:

$y = kx^3$

Step 2: Apply the initial condition $y(1) = -1$

Substitute $x = 1$ and $y = -1$:

$-1 = k(1^3)$

This gives $k = -1$. Therefore, the solution is:

$y = -x^3$

Final Answer:

The solution is (B) $y = -x^3$.

Question 6: Which differential equation matches the slope field?

Here, we analyze the slope field provided and compare it to the given differential equations.

The equations are:

(A) $\frac{dy}{dx} = 10y \left( 1 - \frac{y}{3} \right)$
(B) $\frac{dy}{dx} = \frac{y}{2} \left( 1 - \frac{y}{3} \right)$
(C) $\frac{dy}{dx} = y \left( 1 - \frac{y}{3} \right)$
(D) $\frac{dy}{dx} = 5y \left( 1 - \frac{y}{6} \right)$

Step 1: Identifying key characteristics

The slope field shows that the slopes depend on $y$, not $x$. This eliminates any equations without such dependence.

1. The behavior shows $y = 0$ is an equilibrium solution (horizontal slope).
2. Slopes increase for positive $y$ and decrease for negative $y$, which matches logistic growth.

Step 2: Matching the logistic growth form

The logistic equation has the general form:

$\frac{dy}{dx} = ky \left( 1 - \frac{y}{L} \right)$

where $k$ is a constant and $L$ is the carrying capacity.

From the options:

• (C) matches the form $\frac{dy}{dx} = y \left( 1 - \frac{y}{3} \right)$ with $k = 1$ and $L = 3$.

Final Answer:

The differential equation that matches the slope field is (C).

Summary of Answers:

1. Question 5: (B) $y = -x^3$
2. Question 6: (C) $\frac{dy}{dx} = y \left( 1 - \frac{y}{3} \right)$

Do you want more details on any of the steps? Let me know!

Related Questions:

1. How do you solve separable differential equations?
2. What is a slope field, and how is it related to a differential equation?
3. Can you explain the logistic growth model and its real-life applications?
4. How do equilibrium solutions work in differential equations?
5. What are the general methods for solving first-order linear differential equations?

Tip: In separable differential equations, always check for initial conditions to determine the particular solution!