Picards Theorm Determine the region of the xy plane for whic

Picard\'s Theorm: Determine the region of the xy plane for which dy/dx=sqrt xy and has a unique sol. through (X0,Y0) in the region. How do I set this up and solve using Picards Theorm?

Solution

Picard\'s theorem states that the differential equation dy/dx= sqrt(xy), with boundary condition y(x0)= y0 has a unique solution in a region containing (x0, y0) if sqrt(xy) is continuous in that region and sqrt(xy) is \"Lipschitz\" in y in that region.

hence here we are supposed to tell the region where the function sqrt(xy) is continuous and is Lipschits.

hence this is true when x>=0 and y>=0

(xo,yo) could be any of the values except when x <0 and y < 0

for this problem we are not supposed to solve the differential equation we just need to show where the function fails to conform to the Picard\'s Theorem.