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If y=xlog"{"x/((a+b x))"}" , then show t...

If `y=xlog"{"x/((a+b x))"}"` , then show that `x^3(d^2y)/(dx^2)=(x(dy)/(dx)-y)^2dot`

Text Solution

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Given `y//x=[log x- log (a+bx)].` Therefore,
`(1)/(x)(dy)/(dx)-(1)/(x^(2))y=(1)/(x)-(b)/(a+bx)" [Diff. both sides w.r.t. x]"`
`" or "x(dy)/(dx)-y=(ax)/(a+bx)" (1)"`
Differentiating again w.r.t. x, we get
`(x(d^(2)y)/(dx^(2))+(dy)/(dx))-(dy)/(dx)=(a^(2))/(a+bx)^(2)`
`therefore" "x^(3)(d^(2)y)/(dx^(2))=(a^(2)x^(2))/((a+bx)^(2))=(x(dy)/(dx)-y)^(2)`
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