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Question
If `(a + bx)e^(y/x)` = x then prove that `x(d^2y)/(dx^2) = (a/(a + bx))^2`.
Solution
`y/x = log_e (x/(a + bx))` = loge x – loge (a + bx)
On differentiating with respect to x, we get
`\implies (x dy/dx - y)/x^2 = 1/x - 1/(a + bx) d/dx(a + bx) = 1/x - b/(a + bx)`
`\implies x dy/dx - y = x^2(1/x - b/(a + bx)) = (ax)/(a + bx)`
On differentiating again with respect to x, we get
`\implies x (d^2y)/(dx^2) + dy/dx - dy/dx = ((a + bx)a - ax(b))/(a + bx)^2`
`\implies x (d^2y)/(dx^2) = (a/(a + bx))^2`.
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