1 The function \(y ( x )\) satisfies the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$
where
$$\mathrm { f } ( x , y ) = x + 3 + \sin y$$
and
$$y ( 1 ) = 1$$
- 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.
- Use the formula
$$y _ { r + 1 } = y _ { r - 1 } + 2 h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$
with your answer to part (a), to obtain an approximation to \(y ( 1.2 )\), giving your answer to three decimal places.