The family of curves y = e^{a sin x},where 'a | vedclass

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(A) Given the equation of the family of curves: $y = e^{a \sin x}$.Taking the natural logarithm on both sides,we get: $\log y = a \sin x$.Differentiat

Vedclass · Accepted Answer

$y \log y = 	an x \frac{dy}{dx}$

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(A) Given the equation of the family of curves: $y = e^{a \sin x}$.Taking the natural logarithm on both sides,we get: $\log y = a \sin x$.Differentiating both sides with respect to $x$: $\frac{1}{y} \frac{dy}{dx} = a \cos x$.From the equation $\log y = a \sin x$,we can write $a = \frac{\log y}{\sin x}$.Substituting the value of '$a$' into the differentiated equation:$\frac{1}{y} \frac{dy}{dx} = \left( \frac{\log y}{\sin x} ight) \cos x$.$\frac{1}{y} \frac{dy}{dx} = \log y \cot x$.Rearranging the terms to match the options:$\frac{dy}{dx} = y \log y \cot x$,which can be rewritten as $\frac{dy}{dx} 	an x = y \log y$ or $y \log y = 	an x \frac{dy}{dx}$.

The family of curves $y = e^{a \sin x}$,where '$a$' is an arbitrary constant,is represented by the differential equation:

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