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Please solve the following problems.

1. Y = Sin(mSin^-1x) prove that 1-x² - xy^-1 +m²y = 0

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2. Y = Cosec x + Cot x prove that sin x (d²y ) = y²

dx²

3. If Y = 3cos(log x) + 4sin(log x) show that x²(d²y ) + (dy ) + y = 0

dx² dx

$(i) Given y=sin (m \sin^{- 1} x) \Rightarrow \frac{ⅆ}{ⅆ x} y = \cos (m \sin^{- 1} x) \times \frac{m}{\sqrt{1 - x^{2}}} \Rightarrow y_{1} = \frac{m \cos (m \sin^{- 1} x)}{\sqrt{1 - x^{2}}} \Rightarrow (1 - x^{2}) {y_{1}}^{2} = m^{2} \cos^{2} (m \sin^{- 1} x) \Rightarrow (1 - x^{2}) {y_{1}}^{2} = m^{2} [1 - \sin^{2} (m \sin^{- 1} x)] \Rightarrow (1 - x^{2}) {y_{1}}^{2} = m^{2} (1 - y^{2}) \Rightarrow \frac{ⅆ}{ⅆ x} (1 - x^{2}) {y_{1}}^{2} = m^{2} \frac{ⅆ}{ⅆ x} (1 - y^{2}) \Rightarrow (1 - x^{2}) 2 y_{1} y_{2} - 2 x y_{1}^{2} = - m^{2} 2 y y_{1} \Rightarrow (1 - x^{2}) y_{2} - x y_{1} + m^{2} y = 0 (i i) y = \csc x + \cot x \Rightarrow \frac{ⅆ}{ⅆ x} y = - \csc x \cot x - \csc^{2} x \Rightarrow \frac{ⅆ^{2} y}{d x^{2}} = - (\csc x . - \csc^{2} x + (- \csc x \cot x . \cot x)) - 2 \csc x (- \csc x \cot x) \Rightarrow \frac{ⅆ^{2} y}{d x^{2}} = \csc^{3} x + \csc x \cot^{2} x + \csc^{2} x \cot x \Rightarrow \sin x \frac{ⅆ^{2} y}{d x^{2}} = \sin x \csc x (\csc^{2} x + \cot^{2} x + 2 \csc x \cot x) \Rightarrow \sin x \frac{ⅆ^{2} y}{d x^{2}} = (\csc x + \cot x)^{2} = y^{2} \Rightarrow \sin x \frac{ⅆ^{2} y}{d x^{2}} = y^{2} (i i i) y = 3 \cos (\log x) + 4 \sin (\log x) \Rightarrow \frac{ⅆ}{ⅆ x} y = \frac{- 3 \sin (\log x)}{x} + \frac{4 \cos (\log x)}{x} \Rightarrow \frac{ⅆ^{2} y}{ⅆ x^{2}} = \frac{- \sin (\log x) - 7 \cos (\log x)}{x^{2}} Now given expression x^{2} \frac{ⅆ^{2} y}{ⅆ x^{2}} + x \frac{ⅆ y}{ⅆ x} + y \Rightarrow x^{2} (\frac{- \sin (\log x) - 7 \cos (\log x)}{x^{2}}) + x (\frac{1}{x}) (- 3 \sin (\log x) + 4 \cos (\log x)) + 3 \cos (\log x) + 4 \sin (\log x (\Rightarrow - \sin (\log x) - 7 \cos (\log x) - 3 \sin (\log x) + 4 \cos (\log x) + 3 \cos (\log x) + 4 \sin (\log x) \Rightarrow 0 = R H S$

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