Consider the differential equation dxdy=y−x2 with the initial condition y(0)=1. A student wishes to estimate the value of the solution at x=0.4 before comparing it to the analytic result.
(a) Beginning at (0,1), use two equal sub-intervals of length Δx=0.2 to obtain a numerical estimate of y(0.4). Show all intermediate calculations for each increment of x.
(b) Solve the differential equation analytically and determine the exact value of y(0.4).
(c) Calculate the absolute error between the estimate from part (a) and the exact value from part (b).
(d) Determine the percentage error relative to the exact value and state whether the numerical estimate is an overestimate or an underestimate, justifying your conclusion from the structure of the differential equation.