(a)     Find an integrating factor for this differential equation.

(b)     Solve the differential equation given that $y = 1$ when $x = 1$ , giving your answer in the forms $y = f(x)$ .

(a)     Rewrite the equation in the form

$\frac{{{\text{d}}y}}{{{\text{d}}x}} - \frac{2}{x}y = \frac{{{x^2}}}{{{x^2} + 1}}$     M1A1

Integrating factor $= {{\text{e}}^{\int { - \frac{2}{x}{\text{d}}x} }}$     M1

$= {{\text{e}}^{ - 2\ln x}}$     A1

$= \frac{1}{{{x^2}}}$     A1

Note: Accept $\frac{1}{{{x^3}}}$ as applied to the original equation.

[5 marks]

(b)     Multiplying the equation,

$\frac{1}{{{x^2}}}\frac{{{\text{d}}y}}{{{\text{d}}x}} - \frac{2}{{{x^3}}}y = \frac{1}{{{x^2} + 1}}$     (M1)

$\frac{{\text{d}}}{{{\text{d}}x}}\left( {\frac{y}{{{x^2}}}} \right) = \frac{1}{{{x^2} + 1}}$     (M1)(A1)

$\frac{y}{{{x^2}}} = \int {\frac{{{\text{d}}x}}{{{x^2} + 1}}}$     M1

$= \arctan x + C$     A1

Substitute $x = 1,{\text{ }}y = 1$ .     M1

$1 = \frac{\pi }{4} + C \Rightarrow C = 1 - \frac{\pi }{4}$     A1

$y = {x^2}\left( {\arctan x + 1 - \frac{\pi }{4}} \right)$     A1

[8 marks]

Total [13 marks]

The response to this question was often disappointing. Many candidates were unable to find the integrating factor successfully.