Let y = y(x) be the solution of the differential equation, ((2+sinx)/(y+1))dy/dx = -cosx, y>0, y(0)=1. If y(π)=a and dy/dx at x=π is b, then the ordered pair (a,b) is equal to
Let y = y(x) be the solution of the differential equation, ((2+sinx)/(y+1))dy/dx = -cosx, y>0, y(0)=1. If y(π)=a and dy/dx at x=π is b, then the ordered pair (a,b) is equal to
Option 1 -
(2,1)
Option 2 -
(1,-1)
Option 3 -
(2, 3/2)
Option 4 -
(1,1)
Asked by Shiksha User
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1 Answer
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Correct Option - 4
Detailed Solution:(2+sin x)/ (y+1) dy/dx = -cosx, y>0
⇒ dy/ (y+1) = -cosx/ (2+sinx) dx
By integrating both sides:
ln|y+1| = -ln|2+sinx|+lnK
⇒ y+1 = K/ (2+sinx) (y+1>0)
⇒ y (x) = K/ (2+sinx) - 1
Given y (0)=1 ⇒ 1=K/2-1 ⇒ K=4
So, y (x)=4/ (2+sinx)-1
a=y (π)=1
b=dy/dx|x=π = -cosx/ (y (x)+1)|x=π = 1
So, (a, b)= (1,1)
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