Question

differential equations

Answer 1

That is a separable equation
\[
\frac {dy}{dx}= -y^2\\
-\frac {dy}{y^2} = dx\\
\frac 1 y = x + C\\
y= \frac {1}{x+C}
\]
The rest is easy

Answer 2

y=0 is a solution that does not belong the family\( y=\frac 1 {x+C}\)

Answer 3

For the last part
\[

y(0)=\frac 1 { 0+C}= \frac 1 2\implies C=2\\
y=\frac 1 {x+ 2}
\]
is the answer to the last part.

Answer 4

for a.) y=x^-1, y'=-1/x^2=-y^2
so y<=0? isit coz its negative and 1/(large no.) is actually 0?

Answer 5

thanks for the other parts i appreciate! (:

Answer 6

1/K is not actually equal to a zero for large K. Its limit is zero, but the there is no finite number K so that 1/K=0

Answer 7

i see, got it okay thanks!(: