Differential Equations?

Question

Plot a direction field for the equation
y' = -1 -2y
Based on the direction field, determine the behaviour of y as t --> infinity (as t tends to infinity). If this behaviour depends on the initial value of y as t = 0, describe this dependency.

Answer 1

dy / dt = - (1 + 2y)
- ∫ dy / (1 + 2y) = ∫ dt
- (1/2).log (1 + 2y) = t + k
log (1 + 2y) = - 2 (t + k)
1 + 2y = e^((-2).(t + k))
y = (1/2)[ e^((-2).(t + k)) - 1]
Not sure about the physics of boundary conditions-----will leave that to you please!