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ebbflo

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

  • one year ago
  • one year ago

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

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

    • one year ago
  3. myininaya
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    You just integrate both sides of that equation you have @ebbflo

    • one year ago
  4. myininaya
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    You already have the variables separated

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

    • one year ago
  6. myininaya
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    Is your trouble writing it as an algebraic expression?

    • one year ago
  7. ebbflo
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    I see all of your points and have tried those methods but do not obtain the given answer....

    • one year ago
  8. myininaya
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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)

    • one year ago
  9. myininaya
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    Remember @ebbflo it does say write as an algebraic expression.

    • one year ago
  10. myininaya
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    You get trigonometric expression right after integration

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

    • one year ago
  12. myininaya
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    Also recall tan(arctan(p))=p

    • one year ago
  13. myininaya
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    You will use that too

    • one year ago
  14. ebbflo
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    thanks, I understand all the points you have made

    • one year ago
  15. myininaya
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    What about the solution? Have you got the desired solution?

    • one year ago
  16. myininaya
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    Or do you need more help?

    • one year ago
  17. ebbflo
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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...

    • one year ago
  18. ebbflo
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    @myininaya , you can use Latex so that your math text is more clear

    • one year ago
  19. myininaya
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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

    • one year ago
  20. ebbflo
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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...

    • one year ago
  21. myininaya
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    Was your form algebraic?

    • one year ago
  22. myininaya
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    Can you write what you have?

    • one year ago
  23. ebbflo
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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

    • one year ago
  24. myininaya
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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

    • one year ago
  25. ebbflo
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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...;)

    • one year ago
  26. UnkleRhaukus
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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

    • one year ago
  27. ebbflo
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    @unkle agreed, but it is my preference not to do so when not necessary

    • one year ago
  28. ebbflo
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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

    • one year ago
  29. UnkleRhaukus
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    these are some solution to differential equations ,

    • one year ago
  30. ebbflo
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    in order to make a decision as to what a student may find more clear

    • one year ago
  31. UnkleRhaukus
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    your solution must include as many arbitrary constants as the order of the differential equation

    • one year ago
  32. ebbflo
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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

    • one year ago
  33. UnkleRhaukus
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    well, it can but i just looks awful , these are 'neater'

    • one year ago
  34. UnkleRhaukus
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    looking at (e)

    • one year ago
  35. ebbflo
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    yeah that one can but what about (c)?

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

    • one year ago
  37. UnkleRhaukus
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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

    • one year ago
  38. UnkleRhaukus
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    implicit solutions are fine for DE's

    • 8 months ago
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