A-LevelFurther MathematicsDifferential equations9231/12 · May/June 2017 · Q1011 Marks

It is given that x = t½, where x > 0 and t > 0, and y is a function of x. (i) Show that dy/dx = 2t dy/dt and d²y/dx² = 2(d²y/dt²) + 4t(dy/dt)². (ii) Hence show that the differential equation d²y/dx² (8x + 1/x) dy/dx + 12x²y = 4x²e⁻ˣ² () reduces to the differential equation d²y/dt² - 4 dy/dt + 3y = e⁻ᵗ. (AG) (iii) Find the general solution of (), giving y in terms of x.

(i)Show that dy/dx = 2t dy/dt and d²y/dx² = 2(d²y/dt²) + 4t(dy/dt)².[3]
(ii)Hence show that the differential equation d²y/dx² (8x + 1/x) dy/dx + 12x²y = 4x²e⁻ˣ² (*) reduces to the differential equation d²y/dt² - 4 dy/dt + 3y = e⁻ᵗ. (AG)[1]
(iii)Find the general solution of (*), giving y in terms of x.[7]

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Marking points from the official mark scheme (May/June 2017). The mark or mark type is shown in brackets.

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Topic: Differential equations, A-Level Further Mathematics (syllabus 9231). Structured question, 11 marks.
Source: A-Level Further Mathematics, May/June 2017 Paper 1 Variant 2, question 10.

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