(a) What can you say about a solution of the equation y' = —(1/2)y2 just by looking at the differential equation? A The function y must be strictly decreasing on any interval on which it is defined. A The function y must be strictly increasing on any interval on which it is defined.
0 The function y must be decreasing (or equal to 0) on any interval on which it is deﬁned.
A The function y must be increasing (or equal to 0) on any interval on which it is defined. A The function y must be equal to O on any interval on which it is defined.
J (b) Verify that all members of the family y = 2/(x + C) are solutions of the equation in part (a). 2 . 2 J
y= = y=-—-
x+C (x+C)2
. 2 J 1 2 q 2 1 2
LHS=y=——=-—( )=——y =RHS
(x+C)2 2 x+C 2 (c) Can you think of a solution of the differential equation y' = —(1/2)y2 that is not a member of the family in part (b)? A y = e2X is a solution of y' = _(1/2)y2 that is not a member of the family in part (b).
A Every solution of y' = _(1/2)y2 is a member of the family in part (b).
A y = 2 is a solution of y' = _(1/2)y2 that is not a member of the family in part (b).
= 0 is a solution of r: _ 1 2 that is not a member of the family in part (b).
y y ( / )
A y = x is a solution of y' = _(1/2)y2 that is not a member of the family in part (b).
a! (d) Find a solution of the initial-value problem. Y' = —(1/2)y2 Y(0) = 0-2 x
y— 2+5 3: