`(a+bx)e^(y/x)=x` , Prove that `x^3(d^2y)/(dx^2)=(x(dy)/(dx)-y)^2`

Options

A. `((dy)/(dx)+x)^(2)`
B. `(x(dy)/(dx)-y)^(2)`
C. `((dy)/(dx)-y)^(2)`
D. `(x(dy)/(dx)+y)^(2)`

Answer

B

Explanation

`(y)/(x)=logx-log(a+bx)`
`therefore" "(x(dy)/(dx)-y)/(x^(2))=(1)/(x)-(b)/(b+ax)=(a)/(x(a+bx))`
`therefore" "x(dy)/(dx)-y=(ax)/(a+bx)" (1)"`
Differentiating again, we get
`x(d^(2)y)/(dx^(2))=(a^(2))/((a+bx)^(2))`
`therefore" "x^(3)(d^(2)y)/(dx^(2))=(a^(2)x^(2))/(a+bx)^(2))=((xdy)/(dx)-y)^(2)`