If y= show that d^2y dx^2 +2y dy dx =0 - Vidyadip

Question

If $\text{y}=\cot\text{x}$ show that $\frac{\text{d}^2\text{y}}{\text{dx}^2}+2\text{y}\frac{\text{dy}}{\text{dx}}=0$

✓

Answer

$\text{y}=\cot\text{x}$
Differentiating w.r.t.x
$\Rightarrow\frac{\text{dy}}{\text{dx}}=-\text{cosec}^2\text{x}$
Differentiating w.r.t.x
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}=-[2\text{cosec}\text{ x}(-\text{cosec}^2\times\cot\text{x})]=-2\frac{\text{dy}}{\text{dx}}.\text{y}$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}+2\text{y}\frac{\text{dy}}{\text{dx}}=0$
Hence proved.

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