Iterative/numerical methods

10 A large plantation of area \(20 \mathrm {~km} ^ { 2 }\) is becoming infected with a plant disease. At time \(t\) years the area infected is \(x \mathrm {~km} ^ { 2 }\) and the rate of increase of \(x\) is proportional to the ratio of the area infected to the area not yet infected. When \(t = 0 , x = 1\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 1\).

1. Show that \(x\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 19 x } { 20 - x }$$
2. Solve the differential equation and show that when \(t = 1\) the value of \(x\) satisfies the equation \(x = \mathrm { e } ^ { 0.9 + 0.05 x }\).
3. Use an iterative formula based on the equation in part (b), with an initial value of 2 , to determine \(x\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
4. Calculate the value of \(t\) at which the entire plantation becomes infected.

2. $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } - y ^ { 3 } = 4$$

1. Show that $$x \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = a y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + \left( b y ^ { 2 } + c \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$$ where \(a\), \(b\) and \(c\) are integers to be determined. Given that \(y = 1\) at \(x = 2\)
2. determine the Taylor series expansion for \(y\) in ascending powers of \(( x - 2 )\), up to and including the term in \(( x - 2 ) ^ { 3 }\), giving each coefficient in simplest form.

9. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = y \mathrm { e } ^ { x ^ { 2 } } .$$ It is given that \(y = 0.2\) at \(x = 0\).

1. Use the approximation \(\frac { y _ { 1 } - y _ { 0 } } { h } \approx \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) _ { 0 }\), with \(h = 0.1\), to obtain an estimate of the value of \(y\) at \(x = 0.1\).
2. Use your answer to part (a) and the approximation \(\frac { y _ { 2 } - y _ { 0 } } { 2 h } \approx \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) _ { 1 }\), with \(h = 0.1\), to obtain an estimate of the value of \(y\) at \(x = 0.2\). Gives your answer to 4 decimal places.
   (Total 5 marks)

3 This question concerns the family of differential equations \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 - x ^ { a } y \left( { } ^ { * } \right)\)
where \(a\) is \(- 1,0\) or 1 .

1. Determine and describe geometrically the isoclines of (\textit{) when
   1. \(a = - 1\),
   2. \(a = 0\),
   3. \(a = 1\).
2. In this part of the question \(a = 0\).
   1. Write down the solution to \(( * )\) which passes through the point \(( 0 , b )\) where \(b \neq 1\).
   2. Write down the equation of the asymptote to this solution.
3. In this part of the question \(a = - 1\).
   1. Write down the solution to \(( * )\) which passes through the point \(( c , d )\) where \(c \neq 0\).
   2. Describe the relationship between \(c\) and \(d\) when the solution in part (i) has a stationary point.
4. In this part of the question \(a = 1\).
   1. The standard Runge-Kutta method of order 4 for the solution of the differential equation \(\mathrm { f } ( x , y ) = \frac { \mathrm { d } y } { \mathrm {~d} x }\) is as follows.
      \(k _ { 1 } = h \mathrm { f } \left( x _ { n } , y _ { n } \right)\)
      \(k _ { 2 } = h \mathrm { f } \left( x _ { n } + \frac { h } { 2 } , y _ { n } + \frac { k _ { 1 } } { 2 } \right)\)
      \(k _ { 3 } = h \mathrm { f } \left( x _ { n } + \frac { h } { 2 } , y _ { n } + \frac { k _ { 2 } } { 2 } \right)\)
      \(k _ { 4 } = h \mathrm { f } \left( x _ { n } + h , y _ { n } + k _ { 3 } \right)\)
      \(y _ { n + 1 } = y _ { n } + \frac { 1 } { 6 } \left( k _ { 1 } + 2 k _ { 2 } + 2 k _ { 3 } + k _ { 4 } \right)\).
      Construct a spreadsheet to solve (}) in the case \(x _ { 0 } = 0\) and \(y _ { 0 } = 1.5\). State the formulae you have used in your spreadsheet.
   2. Use your spreadsheet with \(h = 0.05\) to find an approximation to the value of \(y\) when \(x = 1\).
   3. The solution to \(( * )\) in which \(x _ { 0 } = 0\) and \(y _ { 0 } = 1.5\) has a maximum point ( \(r , s\) ) with \(0 < r < 1\). Use your spreadsheet with suitable values of \(h\) to estimate \(r\) to two decimal places. Justify your answer.

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 \(x = 0\).

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.

3 This question explores the family of differential equations \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sqrt { 1 + a x + 2 y }\) for various values of the parameter \(a\). Fig. 3 shows the tangent field in the case \(a = 1\). \begin{figure}[h] \end{figure}

1. (A) Sketch the tangent field in the case \(a = - 2\).
   (B) Explain why the tangent field is not defined for the whole coordinate plane.
   (C) Give an inequality which describes the region in which the tangent field is defined.
   (D) Find a value of \(a\) such that the region for which the tangent field is defined includes the entire \(x\)-axis.
2. (A) For the case \(a = 1\), with \(y = 1\) when \(x = 0\), construct a spreadsheet for the Runge-Kutta method of order 2 with formulae as follows, where \(\mathrm { f } ( x , y ) = \frac { \mathrm { d } y } { \mathrm {~d} x }\). $$\begin{aligned} k _ { 1 } & = h \mathrm { f } \left( x _ { n } , y _ { n } \right) \\ k _ { 2 } & = h \mathrm { f } \left( x _ { n } + h , y _ { n } + k _ { 1 } \right) \\ y _ { n + 1 } & = y _ { n } + \frac { 1 } { 2 } \left( k _ { 1 } + k _ { 2 } \right) \end{aligned}$$ State the formulae you have used in your spreadsheet.
   (B) Use your spreadsheet to obtain the value of \(y\) correct to 4 decimal places when \(x = 1\) for
   • \(h = 0.1\)
     and
   • \(h = 0.05\).
3. (A) For the case \(a = 0\) find the analytical solution that passes through the point ( 0,1 ).
   (B) Verify that the solution in part (iii) (A) is a solution to the differential equation.
   (C) Use the solution in part (iii) (A) to find the value of \(y\) correct to 4 decimal places when \(x = 1\).
4. (A) Verify that \(y = - \frac { a } { 2 } x + \frac { a ^ { 2 } } { 8 } - \frac { 1 } { 2 }\) is a solution for all cases when \(a \leq 0\).
   (B) Show that this is the only straight line solution in these cases. \section*{Copyright Information:} OCR is committed to seeking permission to reproduce all third-party content that it uses in the assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
   OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

1 The function \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = \ln \left( 1 + x ^ { 2 } + y \right)$$ and $$y ( 1 ) = 0.6$$

1. Use the Euler formula $$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with \(h = 0.05\), to obtain an approximation to \(y ( 1.05 )\), giving your answer to four decimal places.
2. Use the improved Euler formula $$y _ { r + 1 } = y _ { r } + \frac { 1 } { 2 } \left( k _ { 1 } + k _ { 2 } \right)$$ where \(k _ { 1 } = h \mathrm { f } \left( x _ { r } , y _ { r } \right)\) and \(k _ { 2 } = h \mathrm { f } \left( x _ { r } + h , y _ { r } + k _ { 1 } \right)\) and \(h = 0.05\), to obtain an approximation to \(y ( 1.05 )\), giving your answer to four decimal places.

2 The function \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } + 3 }$$ and $$y ( 1 ) = 2$$

1. Use the Euler formula $$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with \(h = 0.1\), to obtain an approximation to \(y ( 1.1 )\), giving your answer to four decimal places.
2. Use the improved Euler formula $$y _ { r + 1 } = y _ { r } + \frac { 1 } { 2 } \left( k _ { 1 } + k _ { 2 } \right)$$ where \(k _ { 1 } = h \mathrm { f } \left( x _ { r } , y _ { r } \right)\) and \(k _ { 2 } = h \mathrm { f } \left( x _ { r } + h , y _ { r } + k _ { 1 } \right)\) and \(h = 0.1\), to obtain an approximation to \(y ( 1.1 )\), giving your answer to four decimal places.