What if we have a “final” condition, instead of an initial condition y(tn) = yn, can you design a “backward” Euler’s method from the backward difference formula that uses this final to compute the solution at the same points?

Homework 3.12. As we observe Euler’s method was design using the for-

ward difference formula and the initial condition y(t_o) = y_o to yield a solution

Y1) Y2, • • • 7 Yn

that correspond to the points

ti ₁ t2₎ • • • 1 tn•

What if we have a “final” condition, instead of an initial condition y(t_n) = y_n,

can you design a “backward” Euler’s method from the backward difference

formula that uses this final to compute the solution at the same points?