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.

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Definitions:

Linear Relationship

A type of connection between two variables where the change in one is directly proportional to the change in the other.

Positively Skewed Distribution

A type of distribution where the tail on the right side of the distribution's peak is longer or fatter than the left side, indicating that the bulk of the values lie to the left of the mean.

Distribution

The manner in which a statistical data or a population is spread out or arranged over a range of values.

Empirical Rules

Practical guidelines based on observed data which are used to interpret statistical distributions, notably the 68-95-99.7 (or three-sigma) rule for normal distributions.