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The given differential equation is:

\[
(3xy + 2)dx + x^2dy = 0
\]
with the condition \( y(1) = 1 \).

This is a first-order differential equation, and it can be solved using methods such as separation of variables, an exact equation approach, or integrating factors. First, let's check if it is an exact equation.

For an equation of the form \( M(x, y)dx + N(x, y)dy = 0 \), it is exact if:

\[
\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}
\]

Here, \( M(x, y) = 3xy + 2 \) and \( N(x, y) = x^2 \).

Let's compute the partial derivatives:

\[
\frac{\partial M}{\partial y} = 3x, \quad \frac{\partial N}{\partial x} = 2x
\]

Since \( \frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x} \), this is not an exact equation. We will need to use an integrating factor or attempt another method of solving this.

Let me proceed to solve the equation and apply the initial condition \( y(1) = 1 \).The general solution to the differential equation is:

\[
y(x) = \frac{C_1}{x^2} - \frac{1}{x}
\]

Now, applying the initial condition \( y(1) = 1 \):

\[
1 = C_1 - 1
\]

Solving this gives \( C_1 = 2 \). Therefore, the particular solution to the equation is:

\[
y(x) = \frac{2}{x^2} - \frac{1}{x}
\]

Let me know if you'd like further details or have any questions.

Here are 5 related questions for you:
1. What methods can be used to solve non-exact differential equations?
2. How would the solution change if the initial condition were different?
3. Can this equation be solved using the separation of variables method?
4. What are the applications of such first-order differential equations in real life?
5. How does the integrating factor method help in solving non-exact equations?

**Tip:** Always check if a differential equation is exact before attempting to solve it, as exact equations allow for a direct integration approach.

Learn how to solve the first-order differential equation (3xy + 2)dx + x^2 dy = 0 with the given condition y(1) = 1. This detailed step-by-step solution covers methods such as checking for exactness and applying integrating factors. Suitable for advanced high school or college students.

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