If y= ^ -1 show that (1+x^2) d^2y dx^2 +2x dy dx =0 - Vidyadip

Question

If $\text{y}=\tan^{-1}$ show that $(1+\text{x}^2)\frac{\text{d}^2\text{y}}{\text{dx}^2}+2\text{x}\frac{\text{dy}}{\text{dx}}=0$

✓

Answer

$\text{y}=\tan^{-1}$
Differentiating w.r.t.x, we get
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{1}{1+\text{x}^2}$
$\Rightarrow(1+\text{x}^2)\frac{\text{dy}}{\text{dx}}=1$
Differentiating w.r.t.x, we get
$\Rightarrow(1+\text{x}^2)\frac{\text{d}^2\text{y}}{\text{dx}^2}+2\text{x}\frac{\text{dy}}{\text{dx}}=0$
Hence proved

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