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

Question

If $\text{y}=\sin(\log\text{x})$ prove that $\text{x}^2\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=0$

✓

Answer

Here,
$\text{y}=\sin(\log\text{x})$
Differentiating w.r.t.x, we get
$\frac{\text{dy}}{\text{dx}}=\frac{\cos(\log\text{x})}{\text{x}}$
Differentiating w.r.t.x, we get
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{-\sin(\log\text{x})-\cos(\log\text{x})}{\text{x}^2}$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{-\sin(\log\text{x})}{\text{x}^2}-\frac{\cos(\log\text{x})}{\text{x}^2}{}$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{-\text{y}}{\text{x}^2}-\frac{1}{\text{x}}\times\frac{\text{dy}}{\text{dx}}$
$\Rightarrow\text{x}^2\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=0$

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