Morgan Bernard
01/11/2024 · High School
9. Given that \( \mathrm{y}=\frac{\sin ^{-1} x}{\sqrt{1-x^{2}}} \), prove that \( \left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-3 \times \frac{d y}{d x}=y \quad \) (8marks)
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To prove the equation \( (1-x^2) \frac{d^2 y}{d x^2} - 3 \frac{dy}{dx} = y \), we first find the first and second derivatives of \( y = \frac{\sin^{-1} x}{\sqrt{1-x^2}} \). After substituting these derivatives into the equation and simplifying, we confirm that both sides are equal, thus proving the statement.
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