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

Question

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

✓

Answer

We have, $\text{x}^\text{m}.\text{y}^\text{n}=(\text{x}+\text{y})^{\text{m}+\text{n}}\ \ \dots(\text{i})$
Further, differentiating Eq. (ii) i.e., $\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}$ on both the sides w.r.t. x, we get
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{\text{x}.\frac{\text{dy}}{\text{dx}}-\text{y}.1}{\text{x}^2}$
$=\frac{\text{x}.\frac{\text{y}}{\text{x}}-\text{y}}{\text{x}^2}\Big[\because\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}\Big]$
$=0$
Hence proved.

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