Determine general and particular solutions of first-order differential equations of the form 𝑑𝑦 𝑑𝑥 = 𝑓(𝑥), differential equations of the form 𝑑𝑦 𝑑𝑥 = 𝑔(𝑦) and differential equations of the form 𝑑𝑦 𝑑𝑥 = 𝑓(𝑥)𝑔(𝑦) using separation of variables.

Find the particular solution of the differential equation dy/dx = 2x(y + 1) that passes through the point (0, 2).

A container is being filled with water such that the rate of change of volume with respect to time is proportional to the square root of the current volume. This is modelled by the differential equation shown in the stimulus. (a) Determine the general solution of this differential equation. [2 marks] (b) Given that \(V = 4\) when \(t = 0\), determine the particular solution. Express your answer in the form \(V = f(t)\). [1 mark]

Find the particular solution of the differential equation dy/dx = 3x²y given that y = 2 when x = 0.

A particular solution to the differential equation \(\frac{dy}{dx} = \frac{x\sqrt{1 + y^2}}{y(1 + x^2)}\), where \(x > 0\) and \(y > 0\), passes through the point \((1, \sqrt{3})\). Determine this solution in the form \(y = f(x)\). Leave your answer in simplified form.

A particular solution to the differential equation \(\frac{dy}{dx} = \frac{3x^2}{(x^3 + 8)\sec^2(y)}\), where \(x > 0\) and \(0 < y < \frac{\pi}{2}\), passes through the point \((1, \frac{\pi}{4})\). Determine this solution in the form \(y = f(x)\). Leave your answer in simplified form.

A particular solution to the differential equation \(\frac{dy}{dx} = \frac{2x}{(x^2 - 4)\sin(y)}\), where \(x > 2\) and \(0 < y < \pi\), passes through the point \((3, \frac{\pi}{2})\). Determine this solution in the form \(y = f(x)\). Leave your answer in simplified form.

A particular solution to the differential equation \(\frac{dy}{dx} = \frac{2x\,e^{x^2}}{\sqrt{1 - y^2}}\), where \(x \in \mathbb{R}\) and \(-1 < y < 1\), passes through the point \((0, \frac{1}{2})\). Determine this solution in the form \(y = f(x)\). Express your answer in simplified form.