can anyone help me with this problem?

find the minimum possible value of x^2 + y^2 given that x,y are real numbers such that

xy(x^2 - y^2 ) = x^2 + y^2 , x is not equal to 0.

thanx(Talking)

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- August 11th 2009, 09:50 PMnh149find minimum possible value..
can anyone help me with this problem?

find the minimum possible value of x^2 + y^2 given that x,y are real numbers such that

xy(x^2 - y^2 ) = x^2 + y^2 , x is not equal to 0.

thanx(Talking) - August 12th 2009, 01:02 AMGrandad
Hello nh149I don't know whether this works, but have you tried the substitution , since the expressions are homogeneous?

Then

and becomes

So we need the minimum value of i.e.

Sorry, I've no more time at present to investigate further.

Grandad

Edit: added later

This does indeed give a solution. The value of the expression must be positive, which means or . If you differentiate, and put the result equal to zero, you get a quadratic in , which gives values of in the permissible ranges of

and

Substituting either of these values back gives the minimum value of as exactly , but there's a lot of manipulation of surds along the way. (I've checked this numerically on a spreadsheet, and am pretty sure this is correct.)

Grandad