Answer to Question #122443 in Differential Equations for Joseph Se

Question

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.

Answer 1

$(4-y^2)y'=x^2$

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

on rectangular region $R$ theh there is exist interval $I$ on which unique exists.

We have:

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

$\frac {df}{dy}=\frac {2yx^2}{(4-y^2)^2}$

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

Answer: C

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