Question

If y = `x^3 log(1/x)`, then prove that `x(d^(2)y)/(dx^2) − 2 dy/dx + 3x^2 = 0`

Answer 1

y = x³ log`(1/x)`

y = x3 [1 log 1 − log x]

y = x3 [0 − log x]

y = −x3 log x   ...[x³ log x = −y]

`dy/dx = −[x^3 xx 1/x + log x xx 3x^2]`

x `dy/dx = −[x^3 + 3x^3 log x]`

x `dy/dx = −[x^3 + 3(−y)]`

d.w.r. to x

x × `dy^2/dx^2 + dy/dx = −3x^2 + 3 dy/dx`

`x dy^2/dx^2 − 2dy/dx + 3x^2 = 2`