Let's solve the given differential equation step by step: Given: \[ \frac{dy}{dx} = 3x^2 y^2 \] --- ### **Step 1: Separate the Variables** We want all terms involving \( y \) on one side and all terms involving \( x \) on the other. \[ \frac{dy}{dx} = 3x^2 y^2 \] Divide both sides by \( y^2 \): \[ \frac{1}{y^2}\frac{dy}{dx} = 3x^2 \] Now multiply both sides by \( dx \): \[ \frac{1}{y^2}dy = 3x^2 dx \] --- ### **Step 2: Integrate Both Sides** Integrate both sides with respect to their variables: \[ \int \frac{1}{y^2} dy = \int 3x^2 dx \] Recall that \( \int y^{-2} dy = -y^{-1} \) and \( \int x^2 dx = \frac{x^3}{3} \): \[ \int y^{-2} dy = \int 3x^2 dx \] \[ - y^{-1} = x^3 + C \] --- ### **Step 3: Solve for \( y \) (if required)** Rewrite \( - y^{-1} = x^3 + C \): \[ - \frac{1}{y} = x^3 + C \] Multiply both sides by \( -1 \): \[ \frac{1}{y} = -x^3 - C \] Take the reciprocal: \[ y = \frac{1}{-x^3 - C} \] Or, equivalently: \[ y = -\frac{1}{x^3 + C} \] --- ### **Conclusion** **The general solution to the differential equation** \(\frac{dy}{dx} = 3x^2 y^2\) **is:** \[ \boxed{y = -\frac{1}{x^3 + C}} \] where \( C \) is the constant of integration.