Solve the differential equation. \[ \frac{d y}{d x}=9 x^{2} y^{2} \]
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To solve the differential equation \( \frac{dy}{dx} = 9x^2 y^2 \), we can separate the variables. This means we rearrange the equation to have all \(y\) terms on one side and all \(x\) terms on the other side. Rearranging, we get: \[ \frac{1}{y^2} dy = 9x^2 dx \] Next, we integrate both sides: \[ \int \frac{1}{y^2} dy = \int 9x^2 dx \] The left side integrates to \(-\frac{1}{y}\), and the right side integrates to \(3x^3 + C\), where \(C\) is the constant of integration: \[ -\frac{1}{y} = 3x^3 + C \] Now, we can solve for \(y\): \[ y = -\frac{1}{3x^3 + C} \] This gives us the general solution to the differential equation.
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