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ebbflo

  • 2 years ago

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

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

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

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

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

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

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

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

  8. myininaya
    • 2 years ago
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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)

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

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

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

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

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

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

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

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

  17. ebbflo
    • 2 years ago
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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...

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

  19. myininaya
    • 2 years ago
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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

  20. ebbflo
    • 2 years ago
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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...

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

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

  23. ebbflo
    • 2 years ago
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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

  24. myininaya
    • 2 years ago
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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

  25. ebbflo
    • 2 years ago
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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...;)

  26. UnkleRhaukus
    • 2 years ago
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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

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

  28. ebbflo
    • 2 years ago
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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

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

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

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

  32. ebbflo
    • 2 years ago
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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

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

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

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

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

  37. UnkleRhaukus
    • 2 years ago
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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

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

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