AB5. (a) Using the differential equation, evaluate the derivative at each point, then sketch a short
segment having that slope. For example, at (−1, −1), = 2(−1)((−1)^2 + 1) − 4; draw a
steeply decreasing segment at (−1, −1). Repeat this process at each of the other points. The
result follows.
(b) The differential equation is separable.It is given that f passes through (0,1), so 1 = tan (0^2 + c) and
The solution is f(x) = tan
The particular solution must be differentiable on an interval containing the initial point
(0,1). The tangent function has vertical asymptotes at hence:
(Since x^2 ≥ 0, we ignore the left inequality.)