Hey there Sign in to join this conversationNew here? Join for free

Two different methods give two different answer

Announcements Posted on
Study Help needs new mods! 14-04-2014
Post on TSR and win a prize! Find out more... 10-04-2014
    • Thread Starter
    • 0 followers
    Offline

    ReputationRep:
    I was solving a differential equation and got to the step where I had to integrate. I did it two different methods to get two different answers. I was wondering if anyone knows why one method must not work.

    Method 1

     \int \frac{1}{4y} dy

    let  u = 4y

    therefore:

     \frac{du}{dy} = 4

     \frac{dy}{du} = \frac{1}{4}

     dy = \frac{1}{4} du

    so:

     \int \frac{1}{u} \frac{1}{4} du = \frac{1}{4} \int \frac{1}{u} du

     \frac{1}{4} \ln u = \frac{1} {4} \ln (4y)

    Method 2

     \int \frac{1}{4y} dy

     \frac{1}{4} \int \frac{1}{y}

     \frac{1}{4} \ln y

    As you see both gives different answer but I'm sure both methods are correct.

    Thanks,
    Kieron
    • 4 followers
    Offline

    ReputationRep:
    You missed off the '+c'. Your first answer is equivalent to 1/4*ln(4) + 1/4*ln(y).
    • 3 followers
    Offline

    The key is the "+C", which you forgot to add.

    Notice that if g(x) = f(x) + C then \dfrac{dg}{dx} = \dfrac{df}{dx}, so when you integrate two things then you can end up with two different answers, but they will always only differ by a constant.

    Here, \ln 4y = \ln 4 + ln y, so you get \dfrac{1}{4} \ln 4 + \dfrac{1}{4} \ln y. The \dfrac{1}{4}\ln 4 is a constant, and so (if you add your constants of integration) the two answers are actually the same -- this \dfrac{1}{4}\ln 4 can then be absorbed into the constant of integration in your first method to give \dfrac{1}{4}\ln y + C.
    • 3 followers
    Online

    ReputationRep:
    You've forgotten your constant of integration.
    • Thread Starter
    • 0 followers
    Offline

    ReputationRep:
    (Original post by nuodai)
    The key is the "+C", which you forgot to add.

    Notice that if g(x) = f(x) + C then \dfrac{dg}{dx} = \dfrac{df}{dx}, so when you integrate two things then you can end up with two different answers, but they will always only differ by a constant.

    Here, \ln 4y = \ln 4 + ln y, so you get \dfrac{1}{4} \ln 4 + \dfrac{1}{4} \ln y. The \dfrac{1}{4}\ln 4 is a constant, and so (if you add your constants of integration) the two answers are actually the same -- this \dfrac{1}{4}\ln 4 can then be absorbed into the constant of integration in your first method to give \dfrac{1}{4}\ln y + C.
    I can't believe something so simple drove me crazy. Most of the times the constant doesn't change the answer so I tend to forget to put it.

    Thanks
    • 0 followers
    Offline

    ReputationRep:
    (Original post by Darkening Light)
    I can't believe something so simple drove me crazy. Most of the times the constant doesn't change the answer so I tend to forget to put it.

    Thanks
    Indeed! In regards to forgetting about the constant, try integrating 1/x dx by parts some time, with u = 1/x, v' = 1.
    • 0 followers
    Offline

    ReputationRep:
    the +c my friend!
    i forgot it when integrating arcsin and i got two different answers by two methods aswell.
    not to worry, just an easy mistake

Reply

Submit reply

Register

Thanks for posting! You just need to create an account in order to submit the post
  1. this can't be left blank
    that username has been taken, please choose another Forgotten your password?

    this is what you'll be called on TSR

  2. this can't be left blank
    this email is already registered. Forgotten your password?

    never shared and never spammed

  3. this can't be left blank

    6 characters or longer with both numbers and letters is safer

  4. this can't be left empty
    your full birthday is required
  1. By completing the slider below you agree to The Student Room's terms & conditions and site rules

  2. Slide the button to the right to create your account

    Slide to join now Processing…

    You don't slide that way? No problem.

Updated: May 10, 2012
Article updates
Reputation gems:
You get these gems as you gain rep from other members for making good contributions and giving helpful advice.