3 In this question you are required to consider the family of differential equations \(\frac { d y } { d x } = \frac { y ^ { a } } { x + 1 } - \frac { 1 } { y } ( * )\)
and its solutions. The parameter \(a\) is a real number. You should assume that \(x \geqslant 0\) and \(y > 0\) throughout this question.

1. In this part of the question \(a = 1\).
   1. On the axes in the Printed Answer Booklet
      • Sketch the isocline defined by \(\frac { d y } { d x } = 0\).
2. Shade and label the region in which \(\frac { \mathrm { dy } } { \mathrm { dx } } > 0\).
3. Shade and label the region in which \(\frac { \mathrm { dy } } { \mathrm { dx } } < 0\).
   (ii) For \(b > 0\), find, in terms of \(b\), the solution to \(( * )\) which passes through the point \(( 0 , b )\).
   (iii) Determine
4. The values of \(b > 0\) for which the solution in (ii) has a turning point.
5. The corresponding maximum value of \(y\).
6. Fig. 3.1 and Fig. 3.2 show tangent fields for two distinct but unspecified values of \(a\). In each case a sketch of the solution curve \(y = \mathrm { g } ( x )\) which passes through \(( 0,2 )\) is shown for \(0 \leqslant x \leqslant 0.5\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{43fdb360-0f80-4794-917c-f28b04181fa4-4_656_648_1777_301} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{43fdb360-0f80-4794-917c-f28b04181fa4-4_656_652_1777_1117} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
\end{figure} (i) For the case in Fig. 3.1 suggest a possible value of \(a\).
(ii) For the case in Fig. 3.2 suggest a possible value of \(a\).
(iii) In each case, continue the sketch of the solution curves for \(0.5 \leqslant x \leqslant 5\) in the Printed Answer Booklet.
(iv) State a feature which is present in one of the curves in part (iii) for \(0.5 \leqslant x \leqslant 5\) but not in the other.
7. 1. The Euler method for the solution of the differential equation \(\frac { \mathrm { dy } } { \mathrm { dx } } = \mathrm { f } ( x , y )\) is as follows $$y _ { n + 1 } = y _ { n } + h f \left( x _ { n } , y _ { n } \right)$$ It is given that \(x _ { 0 } = 0\) and \(y _ { 0 } = 2\).
      • Construct a spreadsheet to solve (*) using the Euler method so that the value of \(a\) and the value of \(h\) can be varied, in the case \(x _ { 0 } = 0\) and \(y _ { 0 } = 2\).
8. State the formulae you have used in your spreadsheet.
   [0pt] [3]
   (ii) In this part of the question \(a = 0.1\).
Use your spreadsheet with \(h = 0.1\) to approximate the value of \(y\) when \(x = 3\) for the solution to (*) in which \(y = 2\) when \(x = 0\).
(iii) In this part of the question \(a = - 0.2\). Use your spreadsheet to approximate, to \(\mathbf { 1 }\) decimal place, the \(x\)-coordinate of the local maximum for the solution to (*) in which \(y = 2\) when \(x = 0\).