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Let y(x) be a solution of (1 + e^x)*y' + y*e^x = 1 with y(0) = 2. Which of the following statements is/are true?
- y(-4) = 0
- y(-2) = 0
- y(x) has a critical point in the interval (-1, 0)
- y(x) has no critical point in the interval (-1, 0)
Correct answer: y(-4) = 0
Solution
The equation is exact: d/dx[(1+e^x)y] = 1, giving (1+e^x)y = x + C with C = 4; then y(-4)=0 and the critical point condition y'=0 (i.e. 1 = y e^x) has no root in (-1,0).
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