A2 June 2020 Paper 1 Q10
10
(a) Find the general solution of the differential equation\[\frac{\mathrm{d}y}{\mathrm{d}x} + \frac{2y}{x} = \frac{x + 3}{x(x - 1)(x^2 + 3)} \qquad (x \gt 1)\] [8 marks]
(b) Find the particular solution for which \(y = 0\) when \(x = 3\)
Give your answer in the form \(y = \mathrm{f}(x)\) [2 marks]
| Scheme | Marks | AO |
|---|---|---|
| Selects a method to solve the differential equation by finding an integrating factor. | M1 | 3.1a |
| Obtains the correct integrating factor \(= x^2\) | A1 | 1.1b |
| Multiplies the differential equation by their integrating factor | M1 | 1.1a |
| Correctly integrates their LHS to obtain their \(x^2y\) | A1F | 1.1b |
| Splits their RHS into appropriate partial fractions, including numerators in the correct form. | M1 | 3.1a |
| Obtains partial fractions of the form \(\dfrac{A}{x - 1} + \dfrac{Bx + c}{x^2 + 3}\) with values for \(A, B, C\), with no extra fractions | M1 | 1.1a |
| Integrates \(\dfrac{1}{x^2 + 3}\) to obtain \(k\tan^{-1}\dfrac{x}{\sqrt{3}}\) | B1 | 1.1b |
| Obtains a completely correct expression for the general solution: \(x^2y = \ln(x - 1) + \sqrt{3}\tan^{-1}\dfrac{x}{\sqrt{3}} + c\) Condone omission of \(c\). ACF | A1 | 1.1b |
Typical solution
\[P = \frac{2}{x} \Rightarrow \int P\,\mathrm{d}x = 2\ln x\]Integrating factor \(= \mathrm{e}^{\int P\,\mathrm{d}x} = x^2\)
\[x^2\frac{\mathrm{d}y}{\mathrm{d}x} + 2xy = \frac{x(x + 3)}{(x - 1)(x^2 + 3)}\]\[\frac{\mathrm{d}}{\mathrm{d}x}(x^2y) = \frac{x^2 + 3x}{(x - 1)(x^2 + 3)}\]\[\frac{x^2 + 3x}{(x - 1)(x^2 + 3)} \equiv \frac{A}{x - 1} + \frac{Bx + C}{x^2 + 3}\]\[\equiv \frac{x^2 + 3 + 3x - 3}{(x - 1)(x^2 + 3)} \equiv \frac{1}{x - 1} + \frac{3}{x^2 + 3}\]\[x^2y = \int\left(\frac{1}{x - 1} + \frac{3}{x^2 + 3}\right)\mathrm{d}x\]\[x^2y = \ln(x - 1) + \frac{3}{\sqrt{3}}\tan^{-1}\frac{x}{\sqrt{3}} + c\]or
\[y = \frac{1}{x^2}\left(\ln(x - 1) + \sqrt{3}\tan^{-1}\frac{x}{\sqrt{3}} + c\right)\]Notes
(corrected from the printed mark scheme: the final A1 guidance prints \(xy = \ldots\); the left-hand side is \(x^2y\), as in the typical solution)
| Scheme | Marks | AO |
|---|---|---|
| Substitutes \((3, 0)\) into their general solution, with a constant of integration, and solves to find a value for it. | M1 | 1.1a |
| Obtains the correct solution in the form \(y = \mathrm{f}(x)\). FT their general solution from (a). Condone their correct \(c\) as a decimal. | A1F | 1.1b |
| (10 marks) |