Standard +0.3 This is a straightforward application of the improved Euler method with all formulas provided. Students must substitute given values into the provided formula with f(x,y) = x + √y, requiring careful arithmetic with square roots but no conceptual insight or problem-solving beyond following the algorithm. Slightly above average difficulty due to being Further Maths content and requiring careful numerical work, but remains a standard textbook exercise.
1 The function \(y ( x )\) satisfies the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$
where
$$\mathrm { f } ( x , y ) = x + \sqrt { y }$$
and
$$y ( 3 ) = 4$$
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 \(y ( 3.1 )\), giving your answer to three decimal places.
1 The function $y ( x )$ satisfies the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$
where
$$\mathrm { f } ( x , y ) = x + \sqrt { y }$$
and
$$y ( 3 ) = 4$$
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 $y ( 3.1 )$, giving your answer to three decimal places.
\hfill \mbox{\textit{AQA FP3 2011 Q1 [5]}}