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Y' = [(xy)^2 – xy]/x^2

  1. Jan 29, 2014 #1

    s3a

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    1. The problem statement, all variables and given/known data
    I’m doing a sample exam, and one of the problems is as follows.:
    “Solve the differential equation y' = [(xy)^2 – xy]/x^2.

    Hint: Try a substitution.”

    The teacher’s solution to this problem is attached as TheSolution.jpg.

    2. Relevant equations
    u = xy

    3. The attempt at a solution
    After having looked at the solution, I understand how to do this problem, but I’m not too sure I get how I would have known to make that particular substitution, if I didn’t have the solution already provided for me. Am I just supposed to try and visualize a substitution, without any system certainty, that would have a derivative that works well in conjunction with the substitution, such that I end up with a separable differential equation, or is there a nice, systematic way of dealing with this problem?

    For example, if this differential equation was homogeneous, y = xv would have been a nice substitution that I could rely on to transform the homogeneous, differential equation into a separable one (because that is the systematic way of dealing with homogeneous, differential equations).

    Is there a systematic procedure in the teacher’s solution to this problem that I am missing?

    Any input would be GREATLY appreciated!
     

    Attached Files:

  2. jcsd
  3. Jan 29, 2014 #2

    mfb

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    Staff: Mentor

    That's the basic idea. With experience, the ratio of useful to useless substitution attempts will increase.
     
  4. Jan 29, 2014 #3

    s3a

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    Thanks for the reply.

    Just to confirm, for these kinds of problems, is the substitution always supposed to yield a separable, differential equation, or could it be any other kind of differential equation that I am able to systematically solve?
     
    Last edited: Jan 29, 2014
  5. Jan 30, 2014 #4

    mfb

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    Staff: Mentor

    It can give any other equation type - it can get easy, it can need another substitution to solve, and it can even be worse than the original equation.
     
  6. Jan 30, 2014 #5

    s3a

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    Alright, thanks! :)
     
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