Chapter D1.3, Problem 53E

Analysis of a separable equation Consider the differential equation $y{y}^{\prime }\left(t\right)=\frac{1}{2}{e}^t+t$ and carry out the following analysis.

a.    Find the general solution of the equation and express it explicitly as a function of t in two cases: y > 0 and y < 0.

b.    Find the solutions that satisfy the initial conditions y(–1) = 1 and y(– 1) = 2.

c.    Graph the solutions in part (b) and describe their behavior as t increases.

d.    Find the solutions that satisfy the initial conditions y(–1) = –1 and y(– 1) = –2.

e.    Graph the solutions in part (d) and describe their behavior as t increases.