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Math Help - differential equations

  1. #1
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    differential equations

    solve problem: dy/dx = e^(x+y)... (make final answer e.g. y= something)

    After integrating 1/e^y with respect to dy and e^x with respect to dx, and without including the constant yet, i got
    -e^(-y) = e^x
    where -e^(-y) is from integration of 1/e^y with respect to dy.

    If i write -e^(-y) + C = e^x, I will get final answer as y = -ln (k - e^x), where k = -C, which is the answer from the textbook.

    Also if I write -e^(-y) = e^x + C, I will get the same correct answer.

    1. My question is that.. is the C (constant) the combination of the constants of the integrals of LHS and RHS? And therefore we only add C to either side of the equation and not both?

    2. No matter which side I place the C at for any differential equation problem, will I get the same result?

    And lastly.. must I include a modulus sign after a log or ln? e.g. ln | 2+x | ?
    or just ln (2+x) will do? Although 2+x must be a positive value..?

    Please clear my doubts.. MILLION THANKS!
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  2. #2
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    Hello, custer!

    1. Is the C the combination of the constants of the integrals of LHS and RHS? .Yes!
    And therefore we only add C to either side of the equation and not both? .Right!

    2. No matter which side I place the C for any differential equation problem,
    will I get the same result? . Yes!
    3. And lastly.. must I include a modulus sign after a log or ln?
    e.g. \ln| 2+x | . . . or just \ln(2+x) will do?
    If it is possible for the argument to be negative, absolute values should be used.

    For your example, we should write: . \ln|2+x|

    . . but for \ln|x^2+4| the absolute values are redundant (but not incorrect).

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  3. #3
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    Quote Originally Posted by Soroban View Post
    Hello, custer!



    If it is possible for the argument to be negative, absolute values should be used.

    For your example, we should write: . \ln|2+x|

    . . but for \ln|x^2+4| the absolute values are redundant (but not incorrect).
    I'd suggest that absolute value is only used when the logarithm arises from solving an integral. If the logarithm is a result of re-arranging an equation, absolute value is not used.
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