If x^ m y^ n = ( x + y)^ m + n , prove that d^ 2 y dx^ 2 = 0. - Vidyadip

Question

$\text{If } x^{\text{m }} \text{y}^{\text{n}} = ({x + \text{y)}^{\text{m + n}}}, \text{prove that} \frac{\text{d}^{2}\text{y}}{\text{d}x^{2}} = 0.$

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Answer

$\text{x}^{\text{m}} . \text{y}^{\text{n}} = \text{(x + y)}^{\text{m + n}}$
$\Rightarrow \text{m} \log \text{x + n} \log \text{y = (m + n)} \log \text{(x + y)}$
$\Rightarrow \frac{\text{m}}{\text{x}} + \frac{\text{n}}{\text{y}}.\frac{\text{dy}}{\text{dx}} = \frac{\text{m + n}}{\text{x + y}} \bigg(1 + \frac{\text{dy}}{\text{dx}}\bigg)$
$\Rightarrow \frac{\text{dy}}{\text{dx}} = \frac{\text{y}}{\text{x}} \text{ }\text{ }\text{ }\text{ }\text{ }\dots\text{(i)}$
$\frac{\text{d}^{2}{\text{y}}}{\text{dx}^{2}} = \frac{\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}}{\text{x}^{2}} \text{ }\text{ }\text{ }\text{ }\text{ } \dots\text{(ii) }$
$= \frac{\text{x}{\frac{\text{y}}{\text{x}} -\text{y}}}{\text{x}^{2}} \text{ }\text{ }\text{ }\text{ }\text{ }\dots\text{using (i)}$
= 0

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