Write a differential equation of the form dy/d = ay+b whose all solutions a) app

Question

Write a differential equation of the form dy/d = ay+b whose all solutions a) approach y = 6, b) diverge from y = 6, and sketch the direction field for the differential equation and solve the equation

Explanation / Answer

The differential equation: dy/dt = a*y + b = a*(y + b/a) is a separable equation: dy/(y + b/a) = a dt Integrating: ln(y + b/a) - ln(c) = a*t where ln(c) is the constant of integration. ln((y + b/a)/c) = a*t y(t) = c*exp(a*t) - b/a Now, for this solution to be bounded as t -> infinity, a must be < 0. When a < 0, the exponential term goes to zero as t-> infinity, so: y(t -> infinity) = -b/a for a < 0 We therefore want -b/a = 2 b = -2a Therefore, any differential equation of the form: dy/dt = a*y - 2a, where a < 0 has solutions that approach a limit of y -> 2 as t -> infinity. There are an infinite number of possible solutions that meet the criteria in this question. Examples would be dy/dy = 2 - y (a = -1) dy/dt = 2 - 3y (a = -3) dy/dt = 30a - 15y (a = -15)

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