IIT JEE , AIEEE Forum - Mathematics , Differential Calculus

toshi

if y = log(x/a+bx)^x,prove that x^3 d^2y/dx^2=(x dy/dx-y)^2

deraj_pv

plz...express ur question clearly using the tools.

edison

Problem) If y = log(x/a+bx)^x,prove that x^3 d^2y/dx^2=(x dy/dx-y)^2

Solution) y = log(x/a+bx)^x= x log(x/a+bx)

or dy/dx = log(x/a+bx) + 1

and d²y/d²x = (1/a + b) /(x/a+bx)

so x³ d²y/d²x = x²(x/a+bx)/(x/a+bx) = x² ...........(1)

Now, (x dy/dx-y) = x [log(x/a+bx) + 1] - x log(x/a+bx)

or (x dy/dx-y) = x

or (x dy/dx-y)² = x² ................(2)

Hence from eq. (1) and (2)

x³ d²y/d²x = (x dy/dx-y)² , hence proved.