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Homework Help: First order linear ODE

  1. Jun 7, 2015 #1
    1. The problem statement, all variables and given/known data
    Solve dy/dx = y/x + tan(y/x)

    2. Relevant equations

    3. The attempt at a solution
    Not separable, as far as I can tell. It's not homogeneous, since for the tan term f(λx,λy) = tan(λy/λx) = tan(y/x) ≠ λtan(y/x). It's also not of the form dy/dx + P(x)y = Q(x), because I don't think Q(x) should involve y. And that completes the list of methods I know, none of which I can use! How do you solve this?! Is there a substitution I should make?
  2. jcsd
  3. Jun 7, 2015 #2


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    Science Advisor

    Even though this is not homogeneous, seeing that "y/x" my first thought would be to try the substitution u= y/x.

    Then y= xu so that dy/dx= u+ x du/dx. The differential equation becomes u+ x du/dx= u+ tan(u) so that x du/dx= tan(u).

    That is separable.
  4. Jun 7, 2015 #3


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    Staff Emeritus
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    You could try making a change in the dependent variable, from [itex]y[/itex] to [itex]u[/itex], where [itex]u= y/x[/itex]
  5. Jun 7, 2015 #4
    OK, that substitution works! I thought it was only for homogeneous equations. Thanks :)
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