Solve the differential equation. $$ \frac{d y}{d x}=9 x^{2} y^{2} $$

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We begin with the differential equation

$$
\frac{dy}{dx} = 9x^2 y^2.
$$

**Step 1. Separate the variables.**

We rewrite the equation by dividing both sides by $$ y^2 $$ and multiplying by $$ dx $$:

$$
\frac{1}{y^2} \, dy = 9x^2 \, dx.
$$

**Step 2. Integrate both sides.**

Integrate the left side with respect to $$ y $$ and the right side with respect to $$ x $$:

$$
\int \frac{1}{y^2} \, dy = \int 9x^2 \, dx.
$$

For the left side, we have

$$
\int y^{-2} \, dy = -\frac{1}{y} + C_1.
$$

For the right side, we compute

$$
\int 9x^2 \, dx = 9 \cdot \frac{x^3}{3} = 3x^3 + C_2.
$$

Combine the constants (let $$ C = C_2 - C_1 $$):

$$
-\frac{1}{y} = 3x^3 + C.
$$

**Step 3. Solve for $$ y $$.**

Multiply the entire equation by $$-1$$:

$$
\frac{1}{y} = -3x^3 - C.
$$

Since $$-C$$ is still an arbitrary constant, we can denote it by $$ C $$:

$$
\frac{1}{y} = -3x^3 + C.
$$

Now, take the reciprocal of both sides to solve for $$ y $$:

$$
y = \frac{1}{C - 3x^3}.
$$

Thus, the general solution of the differential equation is

$$
\boxed{y = \frac{1}{C - 3x^3}},
$$

where $$ C $$ is an arbitrary constant.
The solution to the differential equation $$ \frac{dy}{dx} = 9x^{2}y^{2} $$ is $$ y = \frac{1}{C - 3x^{3}} $$, where $$ C $$ is a constant.

Solve the differential equation. \[ \frac{d y}{d x}=9 x^{2} y^{2} \]

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