Results 1 to 7 of 7

Math Help - Having problems with a linear 1st order ODE

  1. #1
    Member
    Joined
    Aug 2010
    Posts
    138

    Having problems with a linear 1st order ODE

    I've got a linear 1st order ODE, which I'm trying to solve via integrating factors:

    \frac{dy}{dx} = 3 - 6x + y - 2xy


    Book Answer:

    y = -3 + ce^{x^2 - x}


    So, from here, this is what I start out with:

    \frac{dy}{dx} + y(-1 + 2x) = 3 - 6x

    P(t) = 2x -1

    U = e^{\int 2x - 1 dx} = e^{x^2 - x}

    \int \frac{d}{dx}[ye^{x^2 - x}] = \int 3e^{x^2 - x} - 6xe^{x^2 - x} dx


    But working from here would seem to require integrating e^{x^2 - x}, which I didn't think was trivially doable.

    Where am I going wrong?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1

    Re: Having problems with a linear 1st order ODE

    Quote Originally Posted by Lancet View Post
    I've got a linear 1st order ODE, which I'm trying to solve via integrating factors:

    \frac{dy}{dx} = 3 - 6x + y - 2xy


    Book Answer:

    y = -3 + ce^{x^2 - x}


    So, from here, this is what I start out with:

    \frac{dy}{dx} + y(-1 + 2x) = 3 - 6x

    P(t) = 2x -1

    U = e^{\int 2x - 1 dx} = e^{x^2 - x}

    \int \frac{d}{dx}[ye^{x^2 - x}] = \int 3e^{x^2 - x} - 6xe^{x^2 - x} dx


    But working from here would seem to require integrating e^{x^2 - x}, which I didn't think was trivially doable.

    Where am I going wrong?
    Put x=x'+2, dx=dx' in the ODE.

    Edit: will not work!
    Last edited by Also sprach Zarathustra; June 14th 2011 at 03:02 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Aug 2010
    Posts
    138

    Re: Having problems with a linear 1st order ODE

    Quote Originally Posted by Also sprach Zarathustra View Post
    Put x=x'+2, dx=dx' in the ODE.
    I don't understand. I also don't get why this can be done...

    Can you clarify, please?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,361
    Thanks
    39

    Re: Having problems with a linear 1st order ODE

    Quote Originally Posted by Lancet View Post
    I've got a linear 1st order ODE, which I'm trying to solve via integrating factors:

    \frac{dy}{dx} = 3 - 6x + y - 2xy


    Book Answer:

    y = -3 + ce^{x^2 - x}


    So, from here, this is what I start out with:

    \frac{dy}{dx} + y(-1 + 2x) = 3 - 6x

    P(t) = 2x -1

    U = e^{\int 2x - 1 dx} = e^{x^2 - x}

    \int \frac{d}{dx}[ye^{x^2 - x}] = \int 3e^{x^2 - x} - 6xe^{x^2 - x} dx


    But working from here would seem to require integrating e^{x^2 - x}, which I didn't think was trivially doable.

    Where am I going wrong?
    If you split up your integral you will have problems. Consider your integral (I have factored out a 3)

    \int 3(1-2x)e^{x^2-x}dx

    and try the substitution u = x^2-x.

    BTW, your initial ODE is separable

    \dfrac{dy}{dx} = 3 - 6x + y - 2xy = (y+3)(1-2x)
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1

    Re: Having problems with a linear 1st order ODE

    Quote Originally Posted by Danny View Post
    If you split up your integral you will have problems. Consider your integral (I have factored out a 3)

    \int 3(1-2x)e^{x^2-x}dx

    and try the substitution u = x^2-x.

    BTW, your initial ODE is separable

    \dfrac{dy}{dx} = 3 - 6x + y - 2xy = (y+3)(1-2x)
    How I didn't see that?!
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member
    Joined
    Aug 2010
    Posts
    138

    Re: Having problems with a linear 1st order ODE

    Quote Originally Posted by Danny View Post

    If you split up your integral you will have problems. Consider your integral (I have factored out a 3)

    Wow, that's a nasty integration example, simple though it is. Thank you for showing that to me!


    Quote Originally Posted by Danny View Post

    BTW, your initial ODE is separable

    <head smack>

    Well, that would have made things considerably easier, had I seen that. Of course, I would have lost out on the integration lesson, so I'm happy with how things turned out.


    Incidentally, is there a trick to spotting what something like this separates out into? Or is it just trial and error?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member
    Joined
    Aug 2010
    Posts
    138

    Re: Having problems with a linear 1st order ODE

    Quote Originally Posted by Also sprach Zarathustra View Post
    How I didn't see that?!
    I had the same reaction.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Linear program with higher order non-linear constraints.
    Posted in the Advanced Math Topics Forum
    Replies: 2
    Last Post: September 12th 2010, 02:36 AM
  2. First Order Linear Differential Equations Problems
    Posted in the Differential Equations Forum
    Replies: 5
    Last Post: September 3rd 2010, 12:01 AM
  3. Replies: 8
    Last Post: June 7th 2010, 02:32 PM
  4. Replies: 4
    Last Post: August 12th 2008, 04:46 AM
  5. Replies: 1
    Last Post: May 11th 2007, 03:01 AM

Search Tags


/mathhelpforum @mathhelpforum