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Arithmetic Progression

If y=log xa...

Question

If $y = log (\frac{x}{a + b x})^{x}$ , prove that $x^{3} \frac{d^{2} y}{d x^{2}} = (x \frac{d y}{d x} - y)^{2}$

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Solution

$y = log {(\frac{x}{a + b x})}^{x} \Rightarrow y = x log (\frac{x}{a + b x})$
$\frac{d y}{d x} = log (\frac{x}{a + b x}) + x \times \frac{(a + b x)}{x} \times [\frac{(a + b x) - b x}{{(a + b x)}^{2}}]$
$\frac{d y}{d x} = \frac{y}{x} + \frac{a + b x - b x}{(a + b x)}$
$\frac{d y}{d x} = \frac{y}{x} + \frac{a}{(a + b x)} \Rightarrow \frac{d^{2} y}{d x^{2}} = ⎡ ⎢ ⎢ ⎢ ⎣ \frac{\frac{d y}{d x} x - y}{x^{2}} ⎤ ⎥ ⎥ ⎥ ⎦ + \frac{d}{d x} [\frac{a}{a + b x}]$
$\Rightarrow \frac{d^{2} y}{d x^{2}} = ⎡ ⎢ ⎢ ⎢ ⎣ \frac{x \frac{d y}{d x} - y}{x^{2}} ⎤ ⎥ ⎥ ⎥ ⎦ - \frac{a b}{{(a + b x)}^{2}}$
from $(1)$ $\frac{(x \frac{d y}{d x} - y)}{x} = \frac{a}{(a + b x)}$
$\frac{d^{2} y}{d x^{2}} = ⎡ ⎢ ⎢ ⎢ ⎣ \frac{x \frac{d y}{d x} - y}{x^{2}} ⎤ ⎥ ⎥ ⎥ ⎦ - ⎡ ⎢ ⎢ ⎢ ⎣ \frac{x \frac{d y}{d x} - y}{x} ⎤ ⎥ ⎥ ⎥ ⎦ \frac{b}{(a + b x)}$
$\frac{d^{2} y}{d x^{2}} = ⎡ ⎢ ⎢ ⎢ ⎣ \frac{x \frac{d y}{d x} - y}{x^{2}} ⎤ ⎥ ⎥ ⎥ ⎦ - ⎡ ⎢ ⎢ ⎢ ⎣ \frac{x \frac{d y}{d x} - y}{x^{2}} ⎤ ⎥ ⎥ ⎥ ⎦ \frac{(b x + a - a)}{(a + b x)}$
$\frac{d^{2} y}{d x^{2}} = ⎡ ⎢ ⎢ ⎢ ⎣ \frac{x \frac{d y}{d x} - y}{x^{2}} ⎤ ⎥ ⎥ ⎥ ⎦ - ⎡ ⎢ ⎢ ⎢ ⎣ \frac{x \frac{d y}{d x} - y}{x^{2}} ⎤ ⎥ ⎥ ⎥ ⎦ \times [1 - \frac{a}{(a + b x)}]$
$\frac{d^{2}}{d x^{2}} = ⎡ ⎢ ⎢ ⎢ ⎣ \frac{x \frac{d y}{d x} - y}{x^{2}} ⎤ ⎥ ⎥ ⎥ ⎦ [1 - 1 + \frac{a}{a + b x}]$
$\frac{d^{2} y}{d x^{2}} = ⎡ ⎢ ⎢ ⎢ ⎣ \frac{x \frac{d y}{d x} - y}{x^{2}} ⎤ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎣ \frac{x \frac{d y}{d x} - y}{x} ⎤ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎣ \frac{a}{(a + b x)} = \frac{x \frac{d y}{d x} - y}{x} ⎤ ⎥ ⎥ ⎥ ⎦$
$\Rightarrow x^{3} \frac{d^{2} y}{d x^{2}} = {(x \frac{d y}{d x} - y)}^{2}$

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Similar questions

y = l n {(\frac{x}{a + b x})}^{x}

x^{3} \frac{d^{2} y}{d x^{2}}

(a + b x) e^{y / x} = x

, then prove that

x^{3} \frac{d^{2} y}{d x^{2}} = {(x \frac{d y}{d x} - y)}^{2}

(a x + b) e^{y / x} = x

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x^{3} \frac{d^{2} y}{d x^{2}} = {(x \frac{d y}{d x} - y)}^{2}

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x^{3} \frac{d^{2} y}{d x^{2}} = {(x \frac{d y}{d x} - y)}^{2}

If $(a x + b) c \frac{y}{x} = x$ , then show that $X^{3} (\frac{d^{2} y}{d x^{2}}) = {(x \frac{d y}{d x} - y)}^{2}$

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