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If y is a function of x and (2+sin x)/(1+y) (dy)/(dx)=-cos x, with initial condition y(0)=1, then the value of y ((π)/(2)) is
- 1/3
- 2/3
- −1/3
- 1
Correct answer: 1/3
Solution
Separating: dy/(1+y) = -cosx/(2+sinx) dx gives ln(1+y) = -ln(2+sinx)+C, so (1+y)(2+sinx) = const. Using y(0)=1: 2*2 = 4. At x=pi/2: (1+y)*3 = 4 -> y = 1/3.
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