Question

if y(x) is a solution of the differential equation ( ( 2+ sin(x )/( 1)+ y ) ) ( d y )/( d x ) = - cos(x) and y(0)=1, then find the value of y(x/2)

Answer 1

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Answer 2

THE VALUE OF$y(\frac{x}{2}) =\frac{4}{2+sin\frac{x}{2}  } - 1$

Step-by-step explanation:

Given ,

$\frac{2 + sinx}{1+y} \frac{dy}{dx} = - cos x$

⇒$\frac{dy}{1+y} =\frac{-cos x dx}{2 + sin x}$

integrating both sides

⇒㏑(1 + y) = -㏑ (2+ sin x) + ㏑ c     [where c is a constant]

⇒㏑(1+y) + ㏑(2+ sin x ) = ㏑c

⇒㏑((1+y)(2+ sin x)) = ㏑c

⇒(1+ y)(2+ sin x) = c

Given y(0) = 1

therefore (1+1 )(2+ sin 0) =c ⇔ c = 4

so (1+y)(2 + sin x) = 4

⇒y (x) = $\frac{4}{2+ sinx} - 1$

The value of $y(\frac{x}{2}) =\frac{4}{2+sin\frac{x}{2} } - 1$