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Question
Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : `e^((x^7 - y^7)/(x^7 + y^7)` = a
Solution
`e^((x^7 - y^7)/(x^7 + y^7)` = a
∴ `(x^7 - y^7)/(x^7 + y^7)` = log a = k ...(Say)
∴ x7 – y7 = kx7 + ky7
∴ (1 + k)y7 = (1 – k)x7
∴ `y^7/x^7 = (1 - k)/(1 + k)`
∴ `y/x = ((1 - k)/(1 + k))^(1/7)`, a constant
Differentiating both sides w.r.t. x, we get
`"d"/"dx"(y/x)` = 0
∴ `(x"dy"/"dx" - y."d"/"dx"(x))/(x^2)` = 0
∴ `x"dy"/"dx" - y xx 1` = 0
∴ `x"dy"/"dx"` = y
∴ `"dy"/"dx" = y/x`.
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