Hello everyone and thanks for looking at my thread,(adsbygoogle = window.adsbygoogle || []).push({});

I had some trouble solving this ODE which was in a textbook by Henry J. Ricardo:

[tex]x(e^y - y') = 2[/tex].

This problem is from a section dealing with linear equations, but there is a hint beside the problem which reads, "Hint: Think of y as the independent variable, x as the dependent variable, and rewrite the equation in terms of dx/dy."

I tried to solve it by doing the following:

[tex]e^y - y' = 2 / x[/tex]

[tex]y' = e^y - 2 / x[/tex]

[tex]y' = (xe^y - 2) / x[/tex]

[tex]dx/dy = x / (xe^y - 2)[/tex]

[tex]xe^y + e^y (dx/dy) = x[/tex]

[tex]dx/dy + x = xe^{-y}[/tex]

[tex]dx/dy = x(e^{-y} - 1)[/tex]

[tex](1 / x)dx = (e^{-y} - 1)dy[/tex]

[tex]ln|x| + C = -e^{-y} - y[/tex]

However, I know that this is not the correct solution; I'm guessing that I went wrong somewhere when I rewrote the equation in terms of [tex]dx/dy[/tex], but I don't know how else I would approach the problem. Where did I go wrong with my work, and how would I proceed to solve the problem correctly?

Thank you very much for your help!

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# Help Solving a Nonlinear ODE

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