Question

Determine a region in the xy-plane for which y'=sqrt(xy) would have a unique solution whose graph passes through a point (xo, yo) in the region. I got xy<0 & y!=0. But the teacher wants me to express the answer as two intervals of x0 and y0.

Answer 1

Hi, Ian!
Note that you can re-write y ' = sqrt(xy) as \[\frac{ dy }{ dx} =\sqrt{x}\sqrt{y},\]
which is a separable differential equation.

Separate this equation so that the y terms are on the left and the x terms on the right.

Integrate both sides, remembering to add a constant of integration at the end.


Determine the domains of Sqrt(x) and Sqrt(y). That will help y ou determine the region on which the differential equation is defined. Arbitrarily choose a point (x0,y0) within this region. Find the constant of integration by substituting the coordinates of this point into your equation relating x and y. Can you complete the problem from this point forward?