If the solution curve y=y(x) of the differential equation (1+y2 ) (1+ ex )dx+x dy=0, x>0 passes through the point (1,1) and y(e)= - ( 32 ) + ( 32 ), then +2 is .
3
Separate the variables: 1+ exxdx+ dy1+y2=0
Integrate: ex+ ( ex)22+ -1y=C
(1, 1): 0+0+ 4=C
x = e: 1+ 12+ -1y= 4 y= ( 4- 32 )= 1- 321+ 32
∴ α = 1, β = 1 ⇒ α + 2β = 3
Answer: 3
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