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jwxie
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Homework Statement
Find the solution of the given initial value problem in explicit form and determine the interval in which the solution is defined.
[itex]\[x dx+ye^{-x}dy = 0\][/itex] with initial condition y(0) = 1
Homework Equations
The Attempt at a Solution
I solved the first part correctly.
The solution is [itex]\[y = \sqrt{[2(1-x)e^{x}-1]}\][/itex] and the book gives the interval -1.68 < x < 0.77
I don't know how to find the interval. I set the expression under the square root greater than or equal to zero. Then I take natural log on both sides...
[itex]\[2e^{x}-2xe^{x}-1 \geq 0\][/itex]
[itex]\[e^{x}-xe^{x} \geq \frac{1}{2}\][/itex]
[itex]\[x-ln(x)+x \geq ln(\frac{1}{2})\][/itex]
and I am lost...
Can anyone help me on this one? Thank you!