Standard +0.3 This is a straightforward application of the improved Euler method with all formulas provided. Students must substitute given values into the formula (f(x,y) = √(2x) + √y, initial condition y(2)=9, h=0.25) and perform one iteration of arithmetic calculations involving square roots. While it requires careful substitution and arithmetic, it's a standard textbook exercise with no problem-solving or conceptual insight required—slightly easier than average due to the single-step nature and explicit formula provision.
1 The function \(y ( x )\) satisfies the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$
where
$$\mathrm { f } ( x , y ) = \sqrt { ( 2 x ) } + \sqrt { y }$$
and
$$y ( 2 ) = 9$$
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 \(h = 0.25\), to obtain an approximation to \(y ( 2.25 )\), giving your answer to two decimal places.
1 The function $y ( x )$ satisfies the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$
where
$$\mathrm { f } ( x , y ) = \sqrt { ( 2 x ) } + \sqrt { y }$$
and
$$y ( 2 ) = 9$$
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 $h = 0.25$, to obtain an approximation to $y ( 2.25 )$, giving your answer to two decimal places.
\hfill \mbox{\textit{AQA FP3 2012 Q1 [5]}}