it would be trivial to deal with if that did not have actually the one in ~ the end because I could use separation that variables. I tried to use a readjust of variables $ y = xi -x$ yet that go not acquire me anywhere. Does a an easy solution to this ODE exist?


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HINT:

Set $displaystylefrac yx=v$

$displaystyleimplies y=vximpliesfracdydx=v+xfracdvdx$

Reference : Homogeneous plain Differential Equation


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$$dfracd ydx=dfracyx+1quadLongrightarrowquadfrac1xdfracd ydx-frac1x^2y=frac1xquadLongrightarrowquadleft(fracyx ight)"-(ln x)"=0quadLongrightarrowquad fracyx=ln x+c,$$for part $c$ constant.

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Cannot fix the non-exact ODE, ns don't know how to find the integrating aspect such the make the precise ODE.
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