1. The problem asks which differential equations can be solved using the method of separation of variables. 2. Separation of variables works when the equation can be written as $$\frac{dy}{dx} = g(x)h(y)$$, meaning the right side is a product of a function of $x$ and a function of $y$. 3. Let's analyze each option: - A: $$\cos(x) \frac{dy}{dx} = \sin(x) \implies \frac{dy}{dx} = \frac{\sin(x)}{\cos(x)} = \tan(x)$$. This is $$\frac{dy}{dx} = g(x)$$ only, no $y$ term, so separable as $$\frac{dy}{dx} = g(x) \cdot 1$$. - B: $$\cos(x) \frac{dy}{dx} = \sin(y) \implies \frac{dy}{dx} = \frac{\sin(y)}{\cos(x)} = g(x)h(y)$$ with $$g(x) = \frac{1}{\cos(x)}$$ and $$h(y) = \sin(y)$$, separable. - C: $$\frac{dy}{dx} = e^{x+y} = e^x e^y$$, product of functions of $x$ and $y$, separable. - D: $$\frac{dy}{dx} = \ln(x + y)$$, cannot be separated into product of functions of $x$ and $y$ alone, not separable. - E: $$\frac{dy}{dx} = e^{2x} \tan(y)$$, product of functions of $x$ and $y$, separable. - F: $$\frac{dy}{dx} = e^{2y} \tan(y)$$, depends only on $y$, no $x$ term, so can be written as $$g(x)h(y)$$ with $$g(x) = 1$$, separable. 4. Therefore, equations A, B, C, E, and F are separable. Final answer: A, B, C, E, F