Differential equations: past paper questions

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• Question 4Oct/Nov 2025, paper 21Structured10 marks

  Find the particular solution of the differential equation $$5\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}+2\frac{\mathrm{d}y}{\mathrm{d}x}+y=x^2+5x+3,$$ given that, when $x=0$, $y=\frac{\mathrm{d}y}{\mathrm{d}x}=0$.

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• Question 5Oct/Nov 2024, paper 21Structured10 marks

  Find the particular solution of the differential equation $$6\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}-5\frac{\mathrm{d}x}{\mathrm{d}t}+x=t^2+t+1,$$ given that, when $t=0$, $x=12$ and $\frac{\mathrm{d}x}{\mathrm{d}t}=-6$.

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• Question 4Oct/Nov 2023, paper 21Structured10 marks

  Find the particular solution of the differential equation $$\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}+2\frac{\mathrm{d}y}{\mathrm{d}x}+3y=27x^2,$$ given that, when $x=0$, $y=2$ and $\frac{\mathrm{d}y}{\mathrm{d}x}=-8$.

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• Question 7Oct/Nov 2024, paper 21Structured10 marks

  Show that an appropriate integrating factor for $$\sqrt{x^2+16}\frac{\mathrm{d}y}{\mathrm{d}x}+y=x\sqrt{x^2+16}$$ is $\frac{1}{4}x+\frac{1}{4}\sqrt{x^2+16}$.

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• Question 5Oct/Nov 2025, paper 22Structured10 marks

  Find the general solution of the differential equation $$\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}+4\frac{\mathrm{d}y}{\mathrm{d}x}+3y=5\cos x\mathrm{.}$$

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• Question 5Oct/Nov 2024, paper 22Structured10 marksHas a diagram

  Find the particular solution of the differential equation $$3\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}+2\frac{\mathrm{d}y}{\mathrm{d}x}+y=x^2,$$ given that, when $x=0$, $y=\frac{\mathrm{d}y}{\mathrm{d}x}=0$.

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• Question 7Oct/Nov 2025, paper 22Structured11 marks

  Show that $\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{2}x\sqrt{4-x^2}+2{\sin }^{-1}\left(\frac{1}{2}x\right))=\sqrt{4-x^2}$.

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• Question 4May/June 2026, paper 23Structured10 marks

  Find the particular solution of the differential equation $$49\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}+14\frac{\mathrm{d}x}{\mathrm{d}t}+x=196{\mathrm{e}}^{\frac{1}{7}t},$$ given that, when $t=0$, $x=7$ and $\frac{\mathrm{d}x}{\mathrm{d}t}=0$.

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• Question 5Oct/Nov 2024, paper 23Structured10 marks

  Find the particular solution of the differential equation $$6\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}-5\frac{\mathrm{d}x}{\mathrm{d}t}+x=t^2+t+1,$$ given that, when $t=0$, $x=12$ and $\frac{\mathrm{d}x}{\mathrm{d}t}=-6$.

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• Question 6May/June 2026, paper 23Structured10 marks

  Show that an appropriate integrating factor for $$(1-x^2)\frac{\mathrm{d}y}{\mathrm{d}x}+y=(1-x^2)\sqrt{1-x}$$ is $\sqrt{\frac{1+x}{1-x}}$.

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• Question 5May/June 2025, paper 21Structured10 marks

  Find the particular solution of the differential equation $$6\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}+3\frac{\mathrm{d}x}{\mathrm{d}t}+6x={\mathrm{e}}^{-t},$$ given that, when $t=0$, $x=\frac{\mathrm{d}x}{\mathrm{d}t}=0$.

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• Question 2May/June 2023, paper 21Structured7 marks

  Use the substitution $z=x+y$ to find the solution of the differential equation $$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1+3x+3y}{3x+3y-1}$$ for which $y=0$ when $x=1$.

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• Question 7May/June 2025, paper 21Structured10 marks

  Find the solution of the differential equation $$\frac{\mathrm{d}y}{\mathrm{d}x}-\frac{x+5}{x^2+10x+61}y=1,$$ given that $y=0$ when $x=3$.

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• Question 7May/June 2024, paper 21Structured12 marks

  Use the substitution $u=1+x^2$ to find $$\int \frac{x}{\sqrt{1+x^2}}\mathrm{d}x\mathrm{.}$$

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• Question 6May/June 2023, paper 21Structured11 marks

  Find the particular solution of the differential equation $$\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}-12\frac{\mathrm{d}x}{\mathrm{d}t}+36x=37\sin t,$$ given that, when $t=0$, $x=\frac{\mathrm{d}x}{\mathrm{d}t}=0$.

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• Question 7May/June 2024, paper 22Structured12 marks

  Use the substitution $u=1+x^2$ to find $$\int \frac{x}{\sqrt{1+x^2}}\mathrm{d}x\mathrm{.}$$

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• Question 6May/June 2023, paper 22Structured11 marks

  Find the particular solution of the differential equation $$\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}-12\frac{\mathrm{d}x}{\mathrm{d}t}+36x=37\sin t,$$ given that, when $t=0$, $x=\frac{\mathrm{d}x}{\mathrm{d}t}=0$.

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• Question 4May/June 2025, paper 23Structured10 marks

  Find the particular solution of the differential equation $$\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}+\frac{\mathrm{d}x}{\mathrm{d}t}-2x=2t^2+t-1,$$ given that, when $t=0$, $x=\frac{\mathrm{d}x}{\mathrm{d}t}=0$.

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• Question 2May/June 2023, paper 23Structured7 marks

  The variables $x$ and $y$ are related by the differential equation $$6\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}+5\frac{\mathrm{d}x}{\mathrm{d}t}+x=t^2+10t+13.$$ Find the general solution for $x$ in terms of $t$.

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• Question 7May/June 2025, paper 23Structured9 marks

  Find the solution of the differential equation $$\frac{\mathrm{d}y}{\mathrm{d}x}-\frac{2x+6}{x^2+6x+5}y=4,$$ given that $y=0$ when $x=0$.

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