In Example 7 we saw that y(x) = 1/(C − x) defines a one-parameter family of solutions of t...

In Example 7 we saw that y(x) = 1/(C − x) defines a one-parameter family of solutions of the differential equation dy/dx = y2.

(a) Determine a value of C so that y(10) = 10.

(b) Is there a value of C such that y(0) = 0? Can you nevertheless find by inspection a solution of dy/dx = y2 such that y(0) = 0?

(c) Figure shows typical graphs of solutions of the form y(x) = 1/(C − x). Does it appear that these solution curves fill the entire xy-plane? Can you conclude that, given any point (a, b) in the plane, the differential equation dy/dx − y2 has exactly one solution y(x) satisfying the condition y(a) = b?