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Unformatted text preview: y (t)2 ) = c
for some constant c. Therefore, each solution curve is contained in a level
curve of ψ .
We claim that each level curve of the function ψ is an ellipse. Indeed,
ψ (x, y ) is a quadratic form in x and y :
ψ (x, y ) = 1
2 a21 a22
a22 −a12 x
y If the quadratic form ψ is positive deﬁnite or negative deﬁnite, then the level
curves of ψ are ellipses. On the other hand, if ψ is indeﬁnite, then the level
curves of ψ will be hyperbolas.
In the present setting, we have
det a21 a22
= −a21 a12 − a2 = −a21 a12 + a11 a22 > 0.
a22 −a12 This shows that the quadratic form ψ (x, y ) is either positive deﬁnite or negative deﬁnite. (It cannot be indeﬁnite since it has positive determinant.)
Therefore, each level curve of ψ is an ellipse.
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