Direction field analysis Consider the general first-order

Chapter 7, Problem 61AE

(choose chapter or problem)

Direction field analysis Consider the general first-order initial value problem \(y^{\prime}(t)=a y+b\), \(y(0)=y_{0}\), for \(t \geq 0\), where a, b, and v0 are real numbers.

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

b. Draw a representative direction field in the case that a > 0. Show that if \(y_{0}>-b / a\), then the solution increases for \(t \geq 0\), and if \(y_{0}<-b / a\), then the solution decreases for \(t \geq 0\).

c. Draw a representative direction field in the case that a > 0. Show that if \(y_{0}>-b / a\), then the solution decreases for \(t \geq 0\), and if \(y_{0}<-b / a\), then the solution increases for \(t \geq 0\).

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