Consider the interpolation nodes {tn-1, tn}. (a) Write down the Lagrange polynomial interpolant P(t) for the function f(t,y(t)) at the nodes above. (b) Consider the IVP y(t) = f(t,y(t)); as in class, integrate both sides from t = tn to t = tn+1. Approximating the resulting right hand side as
f(t,y(t)) dt ~
P(t)dt,
integrate the Lagrange polynomial P(t) from (a) exactly to determine bi and b2 such that
ytn+1-ytn ~ h * f(tn,ytn) + b * f(tn-1,ytn-1),
This is the two-step Adams-Bashforth method. Your answers bi should be given as fractions (not approximate decimals).