1. a. For the differential equation y' = y, find the solution satisfying
y(0) =1.
b. Then use Euler's method with a step size of 0.1 to approximate the value of y(1).
c. Then recompute using step sizes of 0.05 and 0.01. How do these compare to the approximation above with step size 0.1?
d. Using the value of e as approximately 2.718281, how does the error change as we vary the step size?
2. Use Euler's method to approximate the solution of the differential equation y' = (1 + y)/(1 - y) with any initial condition y(0) -:: 1. What happens here?