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Math Help - Finding the general solution of the D.E.

  1. #1
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    Finding the general solution of the D.E.

    Q. Find the general solution of (x+y*e^y)*dy/dx = 1.

    I'm having trouble solving this. I know you have to make this D.E. linear by interchanging dx and dy. So you have to solve for x(y) and treat y as the independent variable and x as the dependent variable, but I don't know how to proceed from here. Any guidance would be greatly appreciated.
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  2. #2
    A Plied Mathematician
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    Try this:

     (x+y e^y)\frac{dy}{dx} = 1

    \frac{dy}{dx}=\frac{1}{x+ye^{y}}

    Can you see what to do from here?
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  3. #3
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    But I do like your idea of treating x and the dependent variable.
    \frac{dx}{dy}= x+ ye^y
    is a fairly straight forward linear equation.
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  4. #4
    Senior Member bugatti79's Avatar
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    Quote Originally Posted by HallsofIvy View Post
    But I do like your idea of treating x and the dependent variable.
    \frac{dx}{dy}= x+ ye^y
    is a fairly straight forward linear equation.
    May I ask is this done via a substitution? I am looking at a math book here which say the method of substitution can be use for 'homogeneous equations', but this is not homogeneous, right?

    Thanks
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  5. #5
    A Plied Mathematician
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    No, it's not homogeneous. But it is linear in x, and hence you can use the standard integrating factor method, as HallsofIvy hinted at.
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