Question

Determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the region.

(25 − y2)y' = x2

Choose the right answer and explain

a. A unique solution exists in the regions y < −5, −5 < y < 5, and y > 5.

b. A unique solution exists in the region y < 5.

c. A unique solution exists in the region consisting of all points in the xy-plane except (0, 5) and (0, −5).

d. A unique solution exists in the region y > −5.

e. A unique solution exists in the entire xy-plane.

Answer 1

23 In the (25-ya) y = x² y'a f (2) y (6) - Yo (25-y2) IVP y = f (x, y, y (ro) - Yo Is fray) and continuous function in some og rectangle x < < ß , Y <y <S containing the point there is a unique solution to contained in Some inferral of are (2o, Y) the Inpm then xo-h <a <noth that is x < x <ß. 25-y2=0 F(x, y) is not continuous cat y=75, and Of continuous at y= = +5. 2. is also not ay -2xy (25-yaga unique Solution at y=5 is no there SO the solution exist in region y <5, -5{y <5 a> 4 cenique and gy5.