The Graph of a Direction Field for the Differential Equation $\frac { d y } { d x } = 1 - y$

question 77

Essay

The graph of a direction field for the differential equation $\frac { d y } { d x } = 1 - y$ is given below: (a) Sketch a solution curve that satisfies the given condition, but without solving the differential equation:
(i) $y ( 0 ) = 1$ (ii) $y ( 2 ) = 1$ (iii) $y ( 0 ) = 0$
(b) Solve the differential equation for each of the conditions in part (a). Compare your answers to the curves you produced in part (a).
(c) What is the relationship between the curves (i) and (ii) in part (a)? Explain why this occurs.