Question

(10.1.7) (a) What can you say about a solution of the equation y' = -y^2 just by looking at the differential equation? (b) Verify that all members of the family y = 1/(x+C) are solutions of the equation in part (a). (c) Can you think of a solution of the differential equation y' = -y^2 that is not a member of the family in part (b)? (d) Find a solution of the initial value problem y' = -y^2; y(0) = 0.5.

Answer 1

(a) I can tell that the solutions are decreasing or stationary throughout their domain because y'<=0.
(b) \(\large y'=\frac{-1}{(x+C)^2}\\
-y^2=\frac{-1}{(x+C)^2}\)
(c) y=0 fits the bill.
(d) We already have the solution from (b). Just plug in x=0 and y=0.5 to get C.