If y=y(x) and \frac{2+\sin x}{y+1}\frac{dy}{dx}=-\cos x, \: y(0)=1, then y(\frac{\pi }{2}) is equal to

  • Option 1)

    \frac{1}{3}

  • Option 2)

    \frac{2}{3}

  • Option 3)

    -\frac{1}{3}

  • Option 4)

    1

 

Answers (1)

Using

Solution of Differential Equation -

\frac{\mathrm{d}y }{\mathrm{d} x} =f\left ( ax+by+c \right )

put

 Z =ax+by+c

 

 

- wherein

Equation with convert to

\int \frac{dz}{bf\left ( z \right )+a} =x+c

 

 

 

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

\int \frac{dy}{1+y} = \int \frac{-\cos x}{2+\sin x}dx\ \ \ \ Let \ 2\times \sin x=t,\ \ \cos xdx=dt

                    =\int \frac{-dt}{t}= -\log t+C

\log \left ( 1+y \right ) = -\log \left ( 2+ \sin x \right )+C\ \ \ \ -(i)

\log \left ( 1+1 \right ) = -\log \left ( 2+ 0 \right )+C= -\log 2+C

\therefore C=\log 4

\therefore \left ( 1+y \right )= \frac{4}{2+\sin x}\ \ \ from \ (i)

1+y = \frac{4}{2+1}= \frac{4}{3}\ \ \left [ \because \sin \frac{\pi }{2} =1\right ]

y = \frac{4}{3} - 1=\frac{1}{3}


Option 1)

\frac{1}{3}

This option is correct

Option 2)

\frac{2}{3}

This option is incorrect

Option 3)

-\frac{1}{3}

This option is incorrect

Option 4)

1

This option is incorrect

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