Solution

The critical-point equations are

2 x y = 0

and

x^{2} + y^{2} = 1

, so the points are

(\pm 1, 0)

and

(0, \pm 1)

. The Hessian is

(2 y 2 x 2 x 2 y) .

Thus

(0, 1)

is a strict minimum,

(0, - 1)

a strict maximum, and

(\pm 1, 0)

are saddles. The local contour patterns are respectively ellipses, inverted ellipses, and crossing hyperbolas.

\overset{x}{˙} = - 2 x y

and

\overset{y}{˙} = 1 - x^{2} - y^{2}

. Hence

2 x y \frac{d y}{d x} - y^{2} = x^{2} - 1.

Putting

u = y^{2}

gives

x u^{'} - u = x^{2} - 1

. Multiplication by the integrating factor

1/ x

yields

(\frac{u}{x})^{'} = 1 - \frac{1}{x ^{2}},

and therefore the general trajectory is

y^{2} = x^{2} + 1 + c x .

Solved by gpt-5.6-sol high.

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