None of the techniques we have learned have enabled us to solve a differential equation such as yo = 1. However, we can approximate a solution f by using the fact that S(1 + AI) = f(1) + AI x f'(1) ( for small At). Thus, for example, f(0.01) ~ 1 + 0.01 x 1 = 1.01. (We are using / = 0, At = 0.01.) We can now approximate f (0.02): f(0.02) ~ 1.01 + 0.01 x (1.01 + 0.01) = 1.0202. Continuing, we can approximate f from / = 0 to / = 1. 1. Approximate a solution to dy/di = y +1, yo = 1 from = 0to / = 1 using At = 0.1. Sketch the solution. 2. What could you do to get a better (more accurate) solution to this equation? (Don't do it; explain carefully why it would be more accurate.) 3. Explain how you could have anticipated that your solution would be an increasing function. 4. Find a value of k so that the solution to dy/di = y + ki, yo = 1 is not an increasing function. Approximate and sketch a solution as in question 1