The image shows a differential equation problem where you are given the equation:

$y(x^2 + y) = C$

and its derived form:

$2xy + (x^2 + 2y)y' = 0.$

You are also given a graph with a notable point at $(0, 2)$, and it seems to represent a family of curves related to the solution of the differential equation.

Let's break it down:

1. The original equation $y(x^2 + y) = C$ represents a family of implicit curves. The constant $C$ represents different members of the family.

2. The second equation is a differential equation obtained by differentiating the first equation implicitly with respect to $x$. Here, $y'$ represents the derivative of $y$ with respect to $x$, i.e., $\frac{dy}{dx}$.

Implicit differentiation process:

Starting with the equation $y(x^2 + y) = C$, we differentiate both sides with respect to $x$: [ \frac{d}{dx} [y(x^2 + y)] = \frac{d}{dx} [C]. ] Since $C$ is constant, its derivative is 0. On the left-hand side, we use the product rule: $\frac{d}{dx} [y(x^2 + y)] = y' (x^2 + y) + y(2x + y').$ Simplifying and rearranging terms: $(y'(x^2 + y) + y(2x + y')) = 0.$ After distributing and combining terms: $y' (x^2 + 2y) + 2xy = 0.$ This is the differential equation presented in the problem.

Next steps:

• You can solve this equation using methods for solving first-order differential equations, or explore the behavior of the solutions graphically by studying the family of curves represented in the plot.

Would you like to proceed with solving the differential equation or explore more about the graph and the solution behavior near the point $(0, 2)$?

Relative Questions:

1. What is the significance of the constant $C$ in the family of curves?
2. How would you approach solving the first-order differential equation derived?
3. What role does implicit differentiation play in deriving the second equation?
4. Can the family of curves be expressed explicitly in terms of $y(x)$?
5. How would you describe the behavior of the solution near $(0, 2)$?

Tip: Implicit differentiation is a powerful tool for handling equations where $y$ and $x$ are intertwined and not easily separable.