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5hassay
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EDIT: my problem is solved, thank you to those who helped
Solve:
[itex]x y^{\prime \prime} = y^{\prime} \log (\frac{y^{\prime}}{x})[/itex]
Note: This is the first part of an undergraduate applications course in differential equations. We were taught to solve second order non-linear equations by doing tricks such as substituting [itex]u = y^{\prime}[/itex], factoring, and then dealing with the two cases that result.
None.
I tried [itex]u := y^{\prime}[/itex], so [itex]u^{\prime} = y^{\prime \prime}[/itex], and so substitution gives
[itex]x u^{\prime} = u \log(\frac{u}{x})[/itex]
but I can't seem to be able to do anything with that.
I also I noticed that
[itex] y^{\prime \prime} = y^{\prime} x^{-1} \log(y^{\prime} x^{-1}) [/itex]
which I thought was interesting. But I couldn't really do anything with it.
Thank you.
Homework Statement
Solve:
[itex]x y^{\prime \prime} = y^{\prime} \log (\frac{y^{\prime}}{x})[/itex]
Note: This is the first part of an undergraduate applications course in differential equations. We were taught to solve second order non-linear equations by doing tricks such as substituting [itex]u = y^{\prime}[/itex], factoring, and then dealing with the two cases that result.
Homework Equations
None.
The Attempt at a Solution
I tried [itex]u := y^{\prime}[/itex], so [itex]u^{\prime} = y^{\prime \prime}[/itex], and so substitution gives
[itex]x u^{\prime} = u \log(\frac{u}{x})[/itex]
but I can't seem to be able to do anything with that.
I also I noticed that
[itex] y^{\prime \prime} = y^{\prime} x^{-1} \log(y^{\prime} x^{-1}) [/itex]
which I thought was interesting. But I couldn't really do anything with it.
Thank you.
Last edited: