Question #122443
1. 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.

(4-y^2 ) y prime = x^2

A. A unique solution exists in the entire xy-plane.
B. A unique solution exists in the regions y < −2, −2 < y < 2, and y > 2.
C. A unique solution exists in the region consisting of all points in the xy-plane except (0, 2) and (0, −2).
D.A unique solution exists in the region y > −2.
E. A unique solution exists in the region y < 2.
1
Expert's answer
2020-06-17T19:10:07-0400

(4y2)y=x2(4-y^2)y'=x^2

According to theorem of Existence of Unique Solution if f(x,y)f(x,y) and df/dydf/dy are continious

on rectangular region RR theh there is exist interval II on which unique exists.


We have:

f(x,y)=x24y2f(x,y)=\frac {x^2}{4-y^2}

dfdy=2yx2(4y2)2\frac {df}{dy}=\frac {2yx^2}{(4-y^2)^2}


f(x,y)f(x,y) and df/dydf/dy are not continious at y=2y=-2 and y=2y=2


Answer: C

A unique solution exists in the region consisting of all points in the xy-plane except (0,2)(0,2) and (0,2)(0,-2)



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