Question

If y V1--<br />+* V1-y=1; prove that​

Answer 1

Step-by-step explanation:

We know that

$\frac{d}{dx} \Bigg( \frac{U}{V} \Bigg) = \frac{V \frac{dU}{dx} - U \frac{dV}{dx} }{ {V}^{2} } \\ \\ y = \frac{ {sin}^{ - 1}x }{ \sqrt{1 - {x}^{2} } } \: \: \: ...eq1\\ \\ \frac{dy}{dx} = \frac{ \sqrt{1 - {x}^{2} } \frac{d( {sin}^{ - 1}x) }{dx} - {sin}^{ - 1}x \frac{d( \sqrt{1 - {x}^{2} } )}{dx} }{ {( \sqrt{1 - {x}^{2} ) }}^{2} } \\ \\ \frac{dy}{dx} = y_1= \frac{ \sqrt{1 - {x}^{2} } \frac{1 }{ \sqrt{1 - {x}^{2} } } - {sin}^{ - 1}x \frac{( - 2x )}{2\sqrt{1 - {x}^{2} } )} }{ {1 - {x}^{2} }} \\ \\ \frac{dy}{dx} = y_1= \frac{ 1 + x \frac{( {sin}^{ - 1}x )}{\sqrt{1 - {x}^{2} } } }{ {1 - {x}^{2} }} \\ \\ \: from \: eq \: 1 \\ \\ y_1= \frac{ 1 + x y }{ {1 - {x}^{2} }} \\ \\cross \: multiply \\ \\ (1 - {x}^{2} )y_1 = 1 + xy \\ \\ (1 - {x}^{2} )y_1-xy = 1$

Hence proved.

Hope it helps you.