Chapter D1.2, Problem 44E

Direction field analysis Consider the first-order initial value problem $y^{\prime }\left(t\right)=ay+b,y\left(0\right)=A$, for t ≥ 0,where a, b, and A are real numbers.

a.    Explain why y = –b/a is an solution and corresponds to a horizontal line in the direction field.

b.    Draw a representative direction fields in the case that a > 0. Show that if A > –b/a, then the solution increases for t ≥ 0 and if A < –b/a, then the solution decreases for t ≥ 0

c.    Draw a representative direction fields in the case that a < 0. Show that if A > –b/a, then the solution decreases for t ≥ 0 and if A < –b/a, then the solution increases for t ≥ 0