11 The variables x and y satisfy the differential equation x² dy/dx + y² + y = 0. It is given that x = 1 when y = 1. (a) Solve the differential equation to obtain an expression for y in terms of x. (b) State what happens to the value of y when x tends to infinity. Give your answer in an exact form.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

1. B1 Correct separation of variables. ∫1/(y²+y) dy = ∫-1/x² dx. Condone missing integral signs or missing dx, dy, but not both.
2. B1 Obtain 1/x
3. *M1 Express 1/(y²+y) in partial fractions or express the denominator of the fraction as a difference of two squares Allow for the correct split of ±1/(y(y+1))
4. A1 Obtain 1/y - 1/(y+1) or 1/(y+½)² - (½)² Allow if coefficients for the partial fractions are correct but followed by an error.
5. A1 Obtain ln y - ln (y+1) Or equivalent, dependent on where they left the minus sign.
6. DM1 Use x=1, y=1 to find constant of integration or as limits in a definite integral in an expression containing terms of the form pln y and qln(1+y) In ½ = 1 + C If they rearrange the equation before finding the constant of integration then the constant must be of the correct form.
7. A1 Correct equation in x and y e.g. ln(y/(1+y)) = 1 - 1/x + ln(1/2)
8. A1 Obtain y = e^(1-1/x)/(2-e^(1-1/x)) Or equivalent e.g. y = 1/(2e^(1-1/x)-1). Accept with decimal value for e^(-1).
9. B1 FT State that y approaches 1/(2e-1) Or exact equivalent. Condone y = 1/(2e-1) FT on an expression in e.