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 .
- Determine and describe geometrically the isoclines of (\textit{) when
- \(a = - 1\),
- \(a = 0\),
- \(a = 1\).
- In this part of the question \(a = 0\).
- Write down the solution to \(( * )\) which passes through the point \(( 0 , b )\) where \(b \neq 1\).
- Write down the equation of the asymptote to this solution.
- In this part of the question \(a = - 1\).
- Write down the solution to \(( * )\) which passes through the point \(( c , d )\) where \(c \neq 0\).
- Describe the relationship between \(c\) and \(d\) when the solution in part (i) has a stationary point.
- In this part of the question \(a = 1\).
- 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. - Use your spreadsheet with \(h = 0.05\) to find an approximation to the value of \(y\) when \(x = 1\).
- 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.