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प्रश्न
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x + y\frac{dy}{dx} = 0\]
|
\[y = \pm \sqrt{a^2 - x^2}\]
|
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उत्तर
We have,
\[y = \pm \sqrt{a^2 - x^2}\]
\[ \Rightarrow y^2 = a^2 - x^2 . . . . . \left( 1 \right)\]
Given differential equation:
\[x + y\frac{dy}{dx} = 0\]
Differentiating both sides of (1) with respect to x, we get
\[2y \frac{dy}{dx} = - 2x\]
\[ \Rightarrow y \frac{dy}{dx} = - x\]
\[ \Rightarrow x + y \frac{dy}{dx} = 0\]
Hence, the given function is the solution to the given differential equation.
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