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ebbflo Group Title

Solve the following D.E., \[\frac{dy}{1+y^2}=\frac{dx}{1+x^2}\], obtaining the result in algebraic form

  • 2 years ago
  • 2 years ago

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  1. ebbflo Group Title
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    the solution given by the text is \[y=x+C(1+xy)\]

    • 2 years ago
  2. lgbasallote Group Title
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    i believe you should cross multiply first \[(1 + x^2)dy - (1+y^2)dx = 0\]

    • 2 years ago
  3. myininaya Group Title
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    You just integrate both sides of that equation you have @ebbflo

    • 2 years ago
  4. myininaya Group Title
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    You already have the variables separated

    • 2 years ago
  5. UnkleRhaukus Group Title
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    \[\int \frac{1}{1+z^2}\text dz=\arctan z+c\]

    • 2 years ago
  6. myininaya Group Title
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    Is your trouble writing it as an algebraic expression?

    • 2 years ago
  7. ebbflo Group Title
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    I see all of your points and have tried those methods but do not obtain the given answer....

    • 2 years ago
  8. myininaya Group Title
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    If so.. Don't forget you only need one constant (and just put it on one side of your equation) Remember the inverse of tan inverse which is tan lol also remember the formula tan(a+b)=(tan(a)+tan(b))/(1-tan(a)*tan(b)) Then remember that tan(constant) is still a constant)

    • 2 years ago
  9. myininaya Group Title
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    Remember @ebbflo it does say write as an algebraic expression.

    • 2 years ago
  10. myininaya Group Title
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    You get trigonometric expression right after integration

    • 2 years ago
  11. myininaya Group Title
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    You want to transform that to an algebraic one using the hint I just gave you

    • 2 years ago
  12. myininaya Group Title
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    Also recall tan(arctan(p))=p

    • 2 years ago
  13. myininaya Group Title
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    You will use that too

    • 2 years ago
  14. ebbflo Group Title
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    thanks, I understand all the points you have made

    • 2 years ago
  15. myininaya Group Title
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    What about the solution? Have you got the desired solution?

    • 2 years ago
  16. myininaya Group Title
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    Or do you need more help?

    • 2 years ago
  17. ebbflo Group Title
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    no, got it, i had the solution, just don't quite understand "why" the book's solution is written in the form it is... the first copyright is 1943, maybe its a style thing...

    • 2 years ago
  18. ebbflo Group Title
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    @myininaya , you can use Latex so that your math text is more clear

    • 2 years ago
  19. myininaya Group Title
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    That one formula is a trig formula The expansion for tan(a+b) let me know if you need anything else

    • 2 years ago
  20. ebbflo Group Title
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    as i said, I had an equivalent solution, I was just curious as to why the book chose to write the solution in the form it did...

    • 2 years ago
  21. myininaya Group Title
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    Was your form algebraic?

    • 2 years ago
  22. myininaya Group Title
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    Can you write what you have?

    • 2 years ago
  23. ebbflo Group Title
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    I have the solution given, I think the book just wanted a form with one arbitrary constant, i generally don't like my solutions to have "y" on the LHS and RHS when it is not absolutely necessary...it was really ore of a "style" question, sorry I should have specified

    • 2 years ago
  24. myininaya Group Title
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    If the equation can be written in the form the book gives, then your answer is right I got the same answer your book got I wrote it in a different form but it is still correct

    • 2 years ago
  25. ebbflo Group Title
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    yes I understand, thank you...as I said I was questioning the style, and now that I think of it it was probably that the books form only take a single line of text whereas mine did not...;)

    • 2 years ago
  26. UnkleRhaukus Group Title
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    differential equations will often have \(y\) on both sides of the solution, but that is alright. we were only trying to get rid of derivatives replace them with constants

    • 2 years ago
  27. ebbflo Group Title
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    @unkle agreed, but it is my preference not to do so when not necessary

    • 2 years ago
  28. ebbflo Group Title
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    I should have been more clear, I was not so much looking for "help" with the problem, but a different perspective on how another might write the solution

    • 2 years ago
  29. UnkleRhaukus Group Title
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    these are some solution to differential equations ,

    • 2 years ago
  30. ebbflo Group Title
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    in order to make a decision as to what a student may find more clear

    • 2 years ago
  31. UnkleRhaukus Group Title
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    your solution must include as many arbitrary constants as the order of the differential equation

    • 2 years ago
  32. ebbflo Group Title
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    exactly uncle, in those solution you provided, the "y" cannot be written explicitly in terms of "x", at least not in a single expression

    • 2 years ago
  33. UnkleRhaukus Group Title
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    well, it can but i just looks awful , these are 'neater'

    • 2 years ago
  34. UnkleRhaukus Group Title
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    looking at (e)

    • 2 years ago
  35. ebbflo Group Title
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    yeah that one can but what about (c)?

    • 2 years ago
  36. ebbflo Group Title
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    (e) more or less is already, not much more to do on that one...

    • 2 years ago
  37. UnkleRhaukus Group Title
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    i think it is clearer to write (c) like \[x = y^2(\ln y + c)\] i dont know why they chose the form they did

    • 2 years ago
  38. UnkleRhaukus Group Title
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    implicit solutions are fine for DE's

    • one year ago
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