1.1. The slopes are positive for y > −1/2, zero (horizontal) for y = −1/2,
and negative for y > −1/2. It follows that the solutions with grow to +∞ if
the initial condition is > −1/2, to −∞, if the initial value is < −1/2, and will
remain constant if the initial value is equal to −1/2.
1.2. Slopes are positive for all y, except for y = 0, when the direction field
is horizontal. It follows that for positive initial values, the solution goes to +∞,
for negative initial values the solutions approach 0, and are constant zero for
zero initial value.
1.3. We have dV /dt = −kR2 , where k is a positive constant, V is volume,
and R is radius. But we have to use only one unknown function, so noting
that V = 4π/3 · R3 , we get the equation dV /dt = −KV 2/3 , where K is some
constant.
1.4. The general solution of the equation is y = Ce−at + b/a. When C =
0, we get the constant b/a function. The solutions are exponential functions
converging to this constant as t goes to +∞. When a increases, the graphs
change their shape. When b increases, they do not change their shape, but
are lifted vertically, when b/a does not change, but both a and b increase, the
constant solution stays the same, but the shape of the exponential functions
changes.
1.5. The integrating factor is µ(t) = e−2t , so that after multiplying by it
we get (ye−2t )0 = t2 . After integrating and dividing by e−2t , we get the general
solution y = t3 e2t /3 + Ce2t . As t → ∞, the solution goes to +∞, independently
of the value of C, since t3 e2t /3 grows faster than e2t .
1.6. We use the method of integrating factors. The integrating factor is
µ(t) = et/2 , hence the equation is equivalent to
(yet/2 )0 = 2et/2 cos t.
Integrating, we get yet/2 = 54 et/2 cos t+ 85 et/2 sin t+C, hence the general solution
is
4
8
y = cos t + sin t + Ce−t/2 .
5
5
Substituting the initial condition y(0) = −1, we get −1 = 4/5 + C, hence
C = − 59 .
Local maximum is found from the equation 0 = y 0 = − 45 sin t + 85 cos t +
9 −t/2
.
10 e
1