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Question
If \[y = \sqrt{x^2 + a^2}\] prove that \[y\frac{dy}{dx} - x = 0\] ?
Solution
\[\text{ We have, y } = \sqrt{x^2 + a^2}\]
Squaring both sides we get,
\[y^2 = x^2 + a^2 \]
\[ \Rightarrow 2y\frac{d y}{d x} = \frac{d}{dx}\left( x^2 + a^2 \right)\]
\[ \Rightarrow 2y\frac{d y}{d x} = \left( 2x \right)\]
\[ \Rightarrow 2y\frac{d y}{d x} = 2x\]
\[ \Rightarrow y\frac{d y}{d x} = x\]
\[ \Rightarrow y\frac{d y}{d x} - x = 0\]
\[\text{ Hence proved }\]
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