Question

Solve: \( y \, dx - (x + 2y^2) \, dy = 0 \).

Hint:

When solving differential equations, rearrange the terms to separate the variables, then integrate to find the general solution.

Answer 1

Step 1: Rearrange the equation: \[ y \, dx = (x + 2y^2) \, dy \] \[ \frac{dx}{dy} = \frac{x + 2y^2}{y} \] Step 2: Separate the variables: \[ \frac{dx}{dy} = \frac{x}{y} + 2y \] Step 3: Integrate both sides: \[ \int \frac{dx}{dy} \, dy = \int \left( \frac{x}{y} + 2y \right) \, dy \] The solution involves simplifying the integrals and then solving for \( x \) and \( y \). Thus, the solution is obtained.