Question

Solve: \( \frac{dy}{dx} = -4xy^2 \).

Hint:

When solving differential equations, always try to separate variables to simplify the equation before integrating.

Answer 1

Step 1: Separate the variables: \[ \frac{dy}{y^2} = -4x \, dx \] Step 2: Integrate both sides: \[ \int \frac{1}{y^2} \, dy = \int -4x \, dx \] The integral of \( \frac{1}{y^2} \) is \( -\frac{1}{y} \), and the integral of \( -4x \) is \( -2x^2 \). Step 3: After integration, we get: \[ -\frac{1}{y} = -2x^2 + C \] Step 4: Solve for \( y \): \[ y = \frac{1}{x^2 + C} \]