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標題:

變分題目、數學高手請進

發問:

2.(a)若 (y^2 – x^2) 隨 [(1/x^2)-(1/y^2)] 而反變，證明 (y^2 – x^2) 正變於x^2y^2。 (b)若 (y^2 + x^2) 隨[(1/x^2)-(1/y^2)] 而反變，證明 (y^4 – x^4) 正變於 x^2y^2。

最佳解答:

2a) let (y^2 - x^2) = k/((1/x^2) - (1/y^2)),k is constant y^2 - x^2 = k/((y^2 - x^2) / x^2y^2) y^2 - x^2 = k(x^2y^2)/(y^2 - x^2) (y^2 - x^2)^2 = k(x^2y^2)<--------(y^2 - x^2)^2 is directly varies to (x^2y^2) y^2 - x^2 = q(xy)<---------------(y^2 - x^2 ) is directly varies to xy 2b) let (y^2 + x^2) = R/{[(1/x^2) - (1/y^2)]},where R is constant (y^2 + x^2) = R(x^2y^2)/(y^2 - x^2) <---------by part a y^4 - x^4 = R(x^2y^2) So, (y^4 - x^4) is directly varies to x^2y^2

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其他解答:

(a) Let (y^2 - x^2) (1/x^2 - 1/y^2)= k (y^2 - x^2) (y^2 - x^2) / (x^2y^2)= k (y^2 - x^2)^2 = kx^2y^2 (y^2 - x^2) = (開方k)xy (y^2 – x^2) 正變於xy (題目打錯) (b) 同上，差唔多