Quick questions on Differential equations: forming, separating variables and modelling - OCR A-Level Maths A

A differential equation links a quantity to its rate of change. The key phrases translate directly: "the rate of change of $y$" is $\dfrac{dy}{dt}$, "proportional to $y$" multiplies by a constant $k$, and a decreasing quantity gives a negative constant.

The general solution contains an arbitrary constant. An initial condition (a known value at a known time) pins it down to a particular solution.

The proportional-rate model $\dfrac{dP}{dt} = kP$ always integrates to exponential growth $P = P_0 e^{kt}$. Two data points determine both $P_0$ and $k$. Recognising this shape lets you go straight to the form and just fit the constants.

Newton's law of cooling says a body cools at a rate proportional to the difference between its temperature and that of its surroundings. The solution always approaches the surrounding temperature $\theta_0$ as $t \to \infty$, because the exponential term decays to zero. Reading off this long-run value is a common final part of a cooling question.

Solve $\dfrac{dy}{dx} = \dfrac{y}{x}$ for $x > 0$. [3 marks]

The number $N$ decays as $\dfrac{dN}{dt} = -0.2N$ with $N = 100$ at $t = 0$. Find $N$ at $t = 5$. [3 marks]