Evaluate the analytical solution of simple first order differential equations with separable variables, including finding particular solutions (separation of variables may require factorisation involving a common factor)

First-order differential equations with separable variables are a foundational topic in A-Level Mathematics (Edexcel), forming a key part of the calculus component. These equations can be written in the form dy/dx = f(x)g(y), where the variables x and y can be separated onto opposite sides of the equation. Solving them involves rearranging to ∫ 1/g(y) dy = ∫ f(x) dx, then integrating both sides. This technique is essential for modelling real-world phenomena such as population growth, radioactive decay, and cooling rates, where the rate of change depends on the current state.

The analytical solution yields a general solution containing an arbitrary constant of integration. To find a particular solution, an initial condition (e.g., y(0) = 1) is used to determine the constant. In some cases, separation requires factorisation to isolate a common factor, such as dy/dx = (x+1)(y-2), where the right-hand side is already factorised. More complex examples may involve factorising an expression like dy/dx = xy + x, which can be rewritten as dy/dx = x(y+1) by factoring out x. Mastery of this topic is crucial for success in the Edexcel A-Level exams, as it frequently appears in both pure mathematics and applied contexts.