The problem you have provided asks to solve a differential equation using the method of separation of variables. The given equation is:

$y' = \frac{x^2}{y(3 - x^2)}$

To solve this, we'll follow the steps of separating the variables $y$ and $x$, and then integrating both sides.

Step 1: Separate the variables

We can rewrite the equation as:

$y \, dy = \frac{x^2 \, dx}{3 - x^2}$

Step 2: Integrate both sides

Now, we integrate both sides of the equation:

$\int y \, dy = \int \frac{x^2 \, dx}{3 - x^2}$

• The left side integrates to:

$\frac{y^2}{2} = \text{Left side integration result}$

• For the right side, we need to integrate $\frac{x^2 \, dx}{3 - x^2}$. To do this, notice that the numerator can be rewritten in terms of the derivative of the denominator. Here's how it can be done:

$\frac{x^2 \, dx}{3 - x^2} = \frac{-1}{2} \int d(3 - x^2) - 3 \int \frac{dx}{3 - x^2}$

This integral can be simplified using a substitution or recognizing a standard form, leading to a logarithmic function. After completing the integration on both sides, we solve for $y$.

Step 3: Solve for $y$

The solution will involve logarithmic functions and an arbitrary constant $C$, and it will take the form:

$y = \sqrt{\ln |3 - x^2| + C}$

Step 4: Identify the Correct Option

Given that your equation result matches the pattern, the correct answer should resemble:

$y = \sqrt{\ln |3 - x^2| + C}$

This matches the first choice given in the image, where the answer is:

$y = \sqrt{\ln |3 - x^2| + C}$

Would you like more details on the integration process or any further questions?

Related Questions:

1. How do you perform the substitution method in integral calculus?
2. What is the method of separation of variables in solving differential equations?
3. Can you explain how to simplify logarithmic expressions after integration?
4. What are some common forms of integrals involving logarithmic functions?
5. How do you determine the constant of integration $C$ in such problems?

Tip:

When solving differential equations using separation of variables, always make sure to check if the variables can indeed be separated and the resulting integrals are solvable using basic techniques.