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Question Number 35244 by JOHNMASANJA last updated on 17/May/18

if  y=((sin^(−1) x)/(1−x^2 ))  show that   (1−x^2 )(dy/dx) −xy=1

$${if}\:\:{y}=\frac{{sin}^{−\mathrm{1}} {x}}{\mathrm{1}−{x}^{\mathrm{2}} }\:\:{show}\:{that}\: \\ $$$$\left(\mathrm{1}−{x}^{\mathrm{2}} \right)\frac{{dy}}{{dx}}\:−{xy}=\mathrm{1} \\ $$

Commented by math1967 last updated on 17/May/18

(1−x^2 )y=sin^(−1) x  (1−x^2 )(dy/dx)−2xy=(1/(√(1−x^2 )))  (1−x^2 )^(3/2) (dy/dx) −2xy=1  But not (1−x^2 )(dy/dx) −2xy=1  ?????????????

$$\left(\mathrm{1}−{x}^{\mathrm{2}} \right){y}=\mathrm{sin}^{−\mathrm{1}} {x} \\ $$$$\left(\mathrm{1}−{x}^{\mathrm{2}} \right)\frac{{dy}}{{dx}}−\mathrm{2}{xy}=\frac{\mathrm{1}}{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }} \\ $$$$\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} \frac{{dy}}{{dx}}\:−\mathrm{2}{xy}=\mathrm{1} \\ $$$${But}\:{not}\:\left(\mathrm{1}−{x}^{\mathrm{2}} \right)\frac{{dy}}{{dx}}\:−\mathrm{2}{xy}=\mathrm{1} \\ $$$$????????????? \\ $$

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