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

DifficultyAllLevel 2Level 3Level 4Level 5
  1. 1
    Find the general solution of dydx=xy\dfrac{dy}{dx} = xy.
    Level 2 · 3 marksTry it
  2. 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 ∘C15\,^\circ\text{C}

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

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

    As it leaves the forge, the horseshoe is at 815 ∘C815\,^\circ\text{C} and is cooling at 60 ∘C60\,^\circ\text{C} per minute.

    Show that

    dθdt=−0.075(θ−15)\frac{d\theta}{dt} = -0.075(\theta - 15)

    (3 marks)
    Level 2 · 3 marksTry it