Determine the interval validity of this separable equation

[jwxie]

Homework Statement

Find the solution of the given initial value problem in explicit form and determine the interval in which the solution is defined.

$\[x dx+ye^{-x}dy = 0\]$ with initial condition y(0) = 1

Homework Equations

The Attempt at a Solution

I solved the first part correctly.

The solution is $\[y = \sqrt{[2(1-x)e^{x}-1]}\]$ 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...
$\[2e^{x}-2xe^{x}-1 \geq 0\]$
$\[e^{x}-xe^{x} \geq \frac{1}{2}\]$
$\[x-ln(x)+x \geq ln(\frac{1}{2})\]$

and I am lost...
Can anyone help me on this one? Thank you!

 

[LCKurtz]

Homework Statement

Find the solution of the given initial value problem in explicit form and determine the interval in which the solution is defined.

$\[x dx+ye^{-x}dy = 0\]$ with initial condition y(0) = 1

Homework Equations

The Attempt at a Solution

I solved the first part correctly.

The solution is $\[y = \sqrt{[2(1-x)e^{x}-1]}\]$ 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...
$\[2e^{x}-2xe^{x}-1 \geq 0\]$
$\[e^{x}-xe^{x} \geq \frac{1}{2}\]$
$\[x-ln(x)+x \geq ln(\frac{1}{2})\]$

and I am lost...
Can anyone help me on this one? Thank you!

Equations like

$\[e^{x}-xe^{x} = \frac{1}{2}\]$

this can only be solved numerically or using the Lambert W function. This is defined as the inverse of xe^x. So if you get your equation in the form y = xe^x, then x = W(y).

There is probably more information than you want to know about this function on the internet. But if you are unfamiliar with this function or don't want to learn about it just now, just solve the equation numerically with a calculator or computer.

 

[jwxie]

Oh right. Thanks. I picked this problem out of the book. I am sure I haven't learned how to solve this yet.
Thanks!

PS: what math course will I learn to solve Lambert W function?
Thanks again!