Which of the following differential equations is separable? A \( \quad x^{2} d y=(2 y+x) d x \) B \( \quad(x+2) \frac{d y}{d x}-y=2 x^{2} \) C \( \quad 3 z d v=\left(2 v^{2}-1\right) d z \) D \( \quad \frac{d y}{d t}+2 t y=10 \)

To determine which of the given differential equations is separable, we need to check if the equation can be written in the form: \[ \frac{dy}{dx} = f(x)g(y) \] where \( f(x) \) and \( g(y) \) are functions of \( x \) and \( y \) respectively. Let's analyze each option: A. \( x^{2} dy = (2y+x) dx \) This equation can be rearranged as: \[ \frac{dy}{dx} = \frac{2y+x}{x^{2}} \] This is not in the separable form. B. \( (x+2) \frac{dy}{dx} - y = 2x^{2} \) Rearranging the equation: \[ \frac{dy}{dx} = \frac{2x^{2} + y}{x+2} \] This is not in the separable form. C. \( 3z dv = (2v^{2}-1) dz \) Rearranging the equation: \[ \frac{dv}{dz} = \frac{2v^{2}-1}{3z} \] This is not in the separable form. D. \( \frac{dy}{dt} + 2ty = 10 \) Rearranging the equation: \[ \frac{dy}{dt} = 10 - 2ty \] This is not in the separable form. Therefore, none of the given differential equations is separable.