1. A multi-step method for solving y' = f(x, y) is given by the backwards difference method: y_{n+1}=\frac{48}{25} y_{n}-\frac{36}{25} y_{n-1}+\frac{16}{25} y_{n-2}-\frac{3}{25} y_{n-3}+\frac{12}{25} h f\left(x_{n+1}, y_{n+1}\right) (a) Find the leading order

part of the truncation error, showing it is O(h^5). (b) For a four-step scheme such as the above method, you might use a power series expansion to find y1, y2 and y3. Consider the differential equation \frac{d y}{d x}=y^{2}+1 \quad \text { with } \quad y(0)=0.1 Find the power series expansion of y(x) up to, and including, the x^5 term, and use this to find y1, y2 and y3, the estimates for y(0.1), y(0.2)and y(0.3).[6 marks] (c) Use the explicit Euler method as a predictor, and three iteration of the above backwards difference method as a corrector, to find estimates of y4 and y5.[8 marks] Note: If you were unable to get estimates for y1, y2 and y3, then you can use the (incorrect) estimates y0 = 0.1, y1 = 0.2, y2 = 0.3 and y3 = 0.4 for this last part.