Question

Draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y at t -> oo. If this behavior depends on the initial value of y at t 0, describe this dependency. (b) y'-2t-1-y.

Answer 1

y' = 2t -1 - y dy/dt = 2t -1 - y d^2y/dt^2 = 2 - dy/dt d^2y/dt^2 = 0 y = 2 dy/dt = 2 - y when y > 2 dy/dt < 0 (decreasing) (y -2) is negative when y > 2 (y -2) is positive dy/dt > 0(increasing) \r\n direction field at t = 0 if initial value at y is greater than 2 it keeps on increasing and if initial value is less than 2 it keeps on decreasing