Differential equations: past paper questions

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• Question 8Oct/Nov 2025, paper 31Structured8 marks

  The variables $x$ and $y$ satisfy the differential equation $$(x^2+1)\frac{\mathrm{d}y}{\mathrm{d}x}=kx{\mathrm{e}}^{2y},$$ where $k$ is a constant. It is given that $y=0$ when $x=0$ and that $y=-\frac{1}{2}$ when $x=1$.

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• Question 10Oct/Nov 2024, paper 31Structured10 marksHas a diagram

  A large cylindrical tank is used to store water. The base of the tank is a circle of radius 4 metres. At time $t$ minutes, the depth of the water in the tank is $h$ metres.

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• Question 7Oct/Nov 2023, paper 31Structured7 marks

  The variables $x$ and $\theta$ satisfy the differential equation $$\frac{x}{\tan \theta }\frac{\mathrm{d}x}{\mathrm{d}\theta }=x^2+3.$$ It is given that $x=1$ when $\theta =0$. Solve the differential equation, obtaining an expression for $x^2$ in terms of $\theta$.

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• Question 10May/June 2026, paper 32Structured11 marks

  Express $\frac{x+3}{(x-1)(2x-6)}$ in partial fractions.

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• Question 10Oct/Nov 2025, paper 32Structured9 marksHas a diagram

  The diagram shows a tank for holding water. The tank is in the shape of a cube of side $50\text{ cm}$. At time $t$ seconds, the depth of water in the tank is $h\text{ cm}$.

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• Question 10Oct/Nov 2024, paper 32Structured13 marks

  A balloon in the shape of a sphere has volume $V$ and radius $r$. Air is pumped into the balloon at a constant rate of $40\pi$ starting when time $t=0$ and $r=0$. At the same time, air begins to flow out of the balloon at a rate of $0.8\pi r$.

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• Question 11Oct/Nov 2023, paper 32Structured9 marks

  The variables $x$ and $y$ satisfy the differential equation $$x^2\frac{\mathrm{d}y}{\mathrm{d}x}+y^2+y=0.$$ It is given that $x=1$ when $y=1$. Solve the differential equation to obtain an expression for $y$ in terms of $x$.

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• Question 10May/June 2026, paper 33Structured8 marks

  The variables $x$ and $y$ satisfy the differential equation $$4{\mathrm{e}}^{2x}\frac{\mathrm{d}y}{\mathrm{d}x}=x(y+3)$$ for $y>-3$. It is given that $y=-2$ when $x=0$. Solve the differential equation and obtain an expression for $y$ in terms of $x$.

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• Question 10Oct/Nov 2025, paper 33Structured8 marksHas a diagram

  Express $\frac{2}{1-9y^2}$ in partial fractions.

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• Question 10Oct/Nov 2024, paper 33Structured8 marks

  A water tank is in the shape of a cuboid with base area $40000{\text{ cm}}^2$. At time $t$ minutes the depth of water in the tank is $h\text{ cm}$. Water is pumped into the tank at a rate of $50000{\text{ cm}}^3$ per minute.

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• Question 8Oct/Nov 2023, paper 33Structured7 marks

  The variables $x$ and $y$ satisfy the differential equation $${\mathrm{e}}^{4x}\frac{\mathrm{d}y}{\mathrm{d}x}={\cos }^{2}3y\mathrm{.}$$ It is given that $y=0$ when $x=2$. Solve the differential equation, obtaining an expression for $y$ in terms of $x$.

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• Question 10May/June 2025, paper 31Structured8 marks

  Find the quotient and remainder when $x^3+5x^2-2x-15$ is divided by $x^2-3$.

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• Question 11May/June 2024, paper 31Structured11 marks

  In a field there are 300 plants of a certain species, all of which can be infected by a particular disease. At time $t$ after the first plant is infected there are $x$ infected plants.

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• Question 10May/June 2025, paper 33Structured10 marksHas a diagram

  The variables $x$ and $y$ satisfy the differential equation $$\sin 4y\frac{\mathrm{d}y}{\mathrm{d}x}=x\sin 2y\sin 3x\mathrm{.}$$ It is given that $y=\frac{1}{12}\pi$ when $x=\frac{1}{2}\pi$. Solve the differential equation, obtaining a relation between $x$ and $y$.

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• Question 9May/June 2024, paper 33Structured11 marksHas a diagram

  A container in the shape of a cuboid has a square base of side $x$ and a height of $(10-x)$. It is given that $x$ varies with time, $t$, where $t>0$. The container decreases in volume at a rate which is inversely proportional to $t$.

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• Question 8May/June 2023, paper 33Structured8 marks

  The variables $x$ and $y$ satisfy the differential equation $$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{y^2+4}{x(y+4)}$$ for $x>0$. It is given that $x=4$ when $y=2\sqrt{3}$. Solve the differential equation to obtain the value of $x$ when $y=2$.

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