Question

ify=xlog(x/a+bx)then prove that x3d2y/dx2=(xdy/dx-y)2

Answer 1

y = x Log [x/(a+bx)]    ---- (1)
Let   x/(a+bx) = z      ---- (2)

To prove  x³ d²y/dx² = (x y' - y)²

y' = Log z + x * (a+bx)/x * a/(a+bx)²
   = Log z + a/(a+bx)   --- (3)

y'' = 1/z * a/(a+bx)² - ab/(a+bx)²

LHS = x³ y'' = a x² / (a+bx) - ab x³/(a+bx)²
         = ax² / (a+bx)² * [a + bx - b x] = a²x² / (a+bx)²

RHS = (x y' - y)²
        = [x Log z + ax/(a+x) - x Log z ]²
        = a²x²/(a+x)²

LHS = RHS Proved.

Answer 2

ANSWER:-----

x Log [x/(a+bx)]    ---- (1)

Let   x/(a+bx) = z      ---- (2)

To prove  x³ d²y/dx² = (x y' - y)²

y' = Log z + x * (a+bx)/x * a/(a+bx)²

   = Log z + a/(a+bx)   --- (3)

y'' = 1/z * a/(a+bx)² - ab/(a+bx)²

LHS = x³ y'' = a x² / (a+bx) - ab x³/(a+bx)²

         = ax² / (a+bx)² * [a + bx - b x] = a²x² / (a+bx)²

RHS = (x y' - y)²

        = [x Log z + ax/(a+x) - x Log z ]²

        = a²x²/(a+x)²

hence prooved:)