Quick questions on Differential equations - TCE Mathematics Specialised (Tasmania)

An equation is separable if it can be written $\tfrac{dy}{dx} = f(x)g(y)$. Rearrange so each variable is with its own differential, then integrate both sides: $$\int \frac{1}{g(y)}\, dy = \int f(x)\, dx.$$

A first order linear equation has the standard form $$\frac{dy}{dx} + P(x)\, y = Q(x).$$

Many Specialist questions are word problems that you must first turn into a differential equation. The pattern is always "rate of change equals something", and the skill is translating the words.

A slope field (direction field) sketches the gradient $\dfrac{dy}{dx} = f(x, y)$ as a short line segment at a grid of points. A solution curve threads through the field tangent to every segment it passes. Slope fields let you see the qualitative behaviour, equilibrium solutions (where $\dfrac{dy}{dx} = 0$) and long-term trends, without solving the equation. An equilibrium is stable if nearby solutions move toward it and unstable if they move away.