3 This question concerns the family of differential equations $$\frac { d y } { d x } = x ^ { 2 } - y + \operatorname { acos } ( x ) \cos ( y ) \quad ( * * )$$ where \(a\) is a constant, \(x \geqslant 0\) and \(y > 0\).

1. In this part of the question \(a = 0\).
   1. Find the solution to (\textbf{) in which \(y = 1\) when \(x = 0\).
   2. In this part of the question \(m\) is a real number. Show that the equation of the isocline \(\frac { \mathrm { dy } } { \mathrm { dx } } = \mathrm { m }\) is a parabola.
   3. Using the result given in part (a)(ii), or otherwise, sketch the tangent field for (}) on the axes in the Printed Answer Booklet.
2. Fig. 3.1 and Fig. 3.2 show the tangent fields for two distinct and unspecified values of \(a\). In each case, a sketch of the solution curve \(\mathrm { y } = \mathrm { g } ( \mathrm { x } )\) which passes through the point \(( 0,2 )\) is shown for \(0 \leqslant x \leqslant \frac { 1 } { 2 }\). \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 3.1} \includegraphics[alt={},max width=\textwidth]{6d485052-b0db-4c33-b374-4fd7b6f0759c-4_399_666_1324_317}
   \end{figure} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 3.2} \includegraphics[alt={},max width=\textwidth]{6d485052-b0db-4c33-b374-4fd7b6f0759c-4_397_661_1324_1192}
   \end{figure}
   1. In each case, continue the sketch of the solution curve for \(\frac { 1 } { 2 } \leqslant x \leqslant 3\) on the axes in the Printed Answer Booklet.
   2. State one feature which is present in the continued solution curve for Fig. 3.1 that is not a feature of the continued solution curve for Fig. 3.2.
   3. Using a slider for \(a\), or otherwise, estimate the value of \(a\) for the solution curve shown in Fig. 3.2.
3. The Euler method for the solution of the differential equation \(\frac { d y } { d x } = f ( x , y )\) is as follows. $$\begin{aligned} & y _ { n + 1 } = y _ { n } + h f \left( x _ { n } , y _ { n } \right)
   & x _ { n + 1 } = x _ { n } + h \end{aligned}$$
   1. Construct a spreadsheet to solve (), so that the value of \(a\) and the value of \(h\) can be varied, in the case \(x _ { 0 = 0\) and \(y _ { 0 } = 1\). State the formulae you have used in your spreadsheet.
   2. In this part of the question \(a = 0\). Use your spreadsheet with \(h = 0.1\) to approximate the value of \(y\) when \(x = 0.5\) for the solution to (}) in which \(y = 1\) when \(x = 0\).
   3. Using part (a)(i), state the accuracy of the approximate value of \(y\) given in part (c)(ii).
   4. State one change to your spreadsheet that could improve the accuracy of the approximate value of \(y\) found in part (c)(ii).
4. The modified Euler method for the solution of the differential equation \(\frac { d y } { d x } = f ( x , y )\) is as follows.
   \(k _ { 1 } = h f \left( x _ { n } , y _ { n } \right)\)
   \(k _ { 2 } = h f \left( x _ { n } + h , y _ { n } + k _ { 1 } \right)\)
   \(y _ { n + 1 } = y _ { n } + \frac { 1 } { 2 } \left( k _ { 1 } + k _ { 2 } \right)\)
   \(\mathrm { x } _ { \mathrm { n } + 1 } = \mathrm { x } _ { \mathrm { n } } + \mathrm { h }\)
   1. Adapt your spreadsheet from part (c)(i) to a spreadsheet to solve (**), so that the value of \(a\) and the value of \(h\) can be varied, in the case \(x _ { 0 } = 0\) and \(y _ { 0 } = 1\). State the formulae you have used in your spreadsheet.
   2. In this part of the question \(a = - 0.5\). Use the spreadsheet from part (d)(i) with \(h = 0.1\) to approximate the value of \(y\) when \(x = 0.5\) for the solution to \(( * * )\) in which \(y = 1\) when \(x = 0\). In this part of the question \(a = - 0.5\). The solution to (**) in which \(y = 1\) when \(x = 0\) has a turning point with coordinates \(( c , d )\) where \(0 < c < 1\).
   3. Use the spreadsheet in part (d)(i) to determine the value of \(c\) correct to \(\mathbf { 1 }\) decimal place.
   4. Use the spreadsheet in part (d)(i) to determine the value of \(d\) correct to \(\mathbf { 3 }\) decimal places.