A-level · Integration

Differential equationsExam-style questions

Solving first-order separable differential equations, using a cooling block of metal throughout. Writing a differential equation from a description of a rate; separating the variables; integrating both sides with one constant; turning a log into an explicit formula; using the conditions to find the constants; and reading the model — its long-term behaviour and its limits, including a logistic model that needs partial fractions.

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57 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
Find the general solution of $\dfrac{dy}{dx} = xy$ .
Level 2 · 3 marksTry it
2
A blacksmith takes a horseshoe out of a forge and leaves it to cool on the stone floor of a workshop. The air in the workshop stays at $15\,^\circ\text{C}$

The temperature of the horseshoe $t$ minutes after it leaves the forge is $\theta\,^\circ\text{C}$

A model of the cooling assumes that the rate at which $\theta$ falls is proportional to $\theta - 15$

As it leaves the forge, the horseshoe is at $815\,^\circ\text{C}$ and is cooling at $60\,^\circ\text{C}$ per minute.

Show that

$\frac{d\theta}{dt} = -0.075(\theta - 15)$

(3 marks)
Level 2 · 3 marksTry it
3
Solve $\dfrac{dy}{dx} = \dfrac{2x}{y}$ given that $y = 3$ when $x = 0$ .
Level 3 · 4 marksTry it
4
Find the general solution of $\dfrac{dy}{dx} = 3x^2 y$ .
Level 3 · 3 marksTry it
5
Solve $\dfrac{dy}{dx} = \dfrac{\cos x}{y}$ given that $y = 2$ when $x = 0$ .
Level 3 · 4 marksTry it

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