Differential equations
Integration
- A2
- Non-calculator
- 8 questions
- 37 marks
These are original practice questions for the Edexcel A-level Pure content, not a past paper. This site is independent and not affiliated with Pearson.
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Find the general solution of $\dfrac{dy}{dx} = 3x^2$.
3 marks
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Find the general solution of $\dfrac{dy}{dx} = y$.
3 marks
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Find the general solution of $\dfrac{dy}{dx} = 3x^2 y$.
4 marks
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Solve $\dfrac{dy}{dx} = 6x^2$, given that $y = 3$ when $x = 1$.
4 marks
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Solve $\dfrac{dy}{dx} = \dfrac{2y}{x}$, given that $y = 3$ when $x = 1$.
5 marks
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The number of insects in a colony is modelled by $\dfrac{dN}{dt} = kN$. When $t = 0$, $N = 100$, and when $t = 5$, $N = 200$. Find $N$ when $t = 10$.
5 marks
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Solve $\dfrac{dy}{dx} = 2xy^2$, given that $y = 1$ when $x = 0$.
6 marks
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The temperature $T$ of an object, in degrees Celsius, satisfies $\dfrac{dT}{dt} = -k(T - 20)$. When $t = 0$, $T = 80$, and when $t = 10$, $T = 50$. Find the temperature when $t = 20$.
7 marks
Mark scheme
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Answer. $y = x^3 + c$
- Integrate both sides with respect to $x$.
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Answer. $y = Ae^{x}$
- Separate the variables: $\dfrac{dy}{y} = dx$.
- $\ln|y| = x + c$, so $y = Ae^x$.
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Answer. $y = Ae^{x^3}$
- Separate: $\dfrac{dy}{y} = 3x^2\,dx$.
- $\ln|y| = x^3 + c$.
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Answer. $y = 2x^3 + 1$
- $y = 2x^3 + c$.
- $3 = 2 + c$, so $c = 1$.
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Answer. $y = 3x^2$
- Separate: $\dfrac{dy}{y} = \dfrac{2}{x}\,dx$.
- $y = Ax^2$.
- The condition gives $A = 3$.
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Answer. $400$
- $N = 100e^{kt}$.
- $200 = 100e^{5k}$, so $k = \dfrac{\ln 2}{5}$.
- When $t = 10$, $N = 100e^{2\ln 2} = 400$.
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Answer. $y = \dfrac{1}{1 - x^2}$
- Separate: $y^{-2}\,dy = 2x\,dx$.
- $-\dfrac{1}{y} = x^2 + c$.
- The condition gives $c = -1$, so $y = \dfrac{1}{1 - x^2}$.
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Answer. $35$
- Separate and integrate to get $T = 20 + Ae^{-kt}$.
- The first condition gives $A = 60$.
- The second gives $e^{-10k} = \dfrac{1}{2}$.
- When $t = 20$, $e^{-20k} = \dfrac{1}{4}$, so $T = 20 + 15 = 35$.