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# Do 15, and use Theorem 1 existence and uniqueness of solutions of section: slope fields and soluton curves. Transcribed Image Text: A more detailed version

Do 15, and use Theorem 1 existence and uniqueness of solutions of section: slope fields and soluton curves. Transcribed Image Text: A more detailed version of Theorem 1 says that, if the function
f (x, y) is continuous near the point (a, b), then at least one so-
lution of the differential equation y’ f(x, y) exists on some
open interval I containing the point x a and, moreover, that
if in addition the partial derivative af/ay is continuous near
(a,b), then this solution is unique on some (perhaps smaller)
interval J. In Problems 11 through 20, determine whether ex-
istence of at least one solution of the given initial value prob-
lem is thereby guaranteed and, if so, whether uniqueness of
that solution is guaranteed.
%3D
dy
11.
dx
2×2 y2; y(1) = -1
dy
12.
x In y;
dx
y(1) = 1
13.
dx
dy
= Vy; y(0) = 1
14.
dx
dy
Vy: y(0) = 0
dy
15.
VX- y; y(2) = 2
dx
dy
16.
dx
X- y; y(2) = 1
dy
17. y
=x-1; y(0) = 1
dx
18.
=x-1: y(1) =0
dx
dy
– In(1
2
19.

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