Results 1 to 5 of 5
Like Tree5Thanks
  • 2 Post By greg1313
  • 1 Post By Cervesa
  • 2 Post By MarkFL

Thread: integration of e^x with function in x

  1. #1
    Member
    Joined
    Aug 2011
    Posts
    123

    integration of e^x with function in x

    Sir,

    while solving a differential equation problem

    I got the equation as y/(x+1) = integration of e3x(x + 1) + c (integration constant)


    what I did in the integration I multiplied and divided by 3


    3e3x(x + 1)
    _________

    3


    e3x(3x + 3) I took f(x) = 3x and f'(x) = 3]
    _________

    3

    I used integration of ex [f(x) + f'(x)]

    = ex f(x) + c

    = xex + c
    ---
    3

    But they have solved the problem by using parts

    They get a different answer.


    e3x(x + 2)
    _________ + c

    9

    Kindly explain were I went wrong?

    with regards,

    Aranga
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Dec 2016
    From
    Earth
    Posts
    261
    Thanks
    139

    Re: integration of e^x with function in x

    Quote Originally Posted by arangu1508 View Post
    Sir,

    while solving a differential equation problem

    I got the equation as y/(x+1) = integration of e3x(x + 1) + c (integration constant)
    .
    .
    .
    I would like you to post the complete original problem, before you started working on it, because it sounds like you gave
    us your version of part of the problem along the way without the original problem statement.
    Thanks from topsquark and arangu1508
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Aug 2011
    Posts
    123

    Re: integration of e^x with function in x

    solve the differential equation

    (x + 1)y' - y = [e^(3x)] [(x + 1)^3]
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member Cervesa's Avatar
    Joined
    Dec 2014
    From
    USA
    Posts
    66
    Thanks
    53

    Re: integration of e^x with function in x

    Quote Originally Posted by arangu1508 View Post
    Sir,

    while solving a differential equation problem

    I got the equation as y/(x+1) = integration of e3x(x + 1) + c (integration constant)


    what I did in the integration I multiplied and divided by 3


    3e3x(x + 1)
    _________

    3


    e3x(3x + 3) I took f(x) = 3x and f'(x) = 3]
    _________

    3

    I used integration of ex [f(x) + f'(x)]

    wouldn't this be

    $\color{red} {e^{f(x)} [f(x) + f'(x)]}$ ?
    you would still need to integrate by parts, no?
    Thanks from arangu1508
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    2,313
    Thanks
    1032

    Re: integration of e^x with function in x

    Quote Originally Posted by arangu1508 View Post
    solve the differential equation

    (x + 1)y' - y = [e^(3x)] [(x + 1)^3]
    I'd begin by writing:

    $\displaystyle y'-\frac{1}{x+1}y=e^{3x}(x+1)^2$

    Compute the integrating factor:

    $\displaystyle \mu(x)=\exp\left(-\int\frac{1}{x+1}\,dx\right)=\frac{1}{x+1}$

    And the ODE becomes:

    $\displaystyle \frac{1}{x+1}y'-\frac{1}{(x+1)^2}y=e^{3x}(x+1)$

    Or:

    $\displaystyle \frac{d}{dx}\left(\frac{y}{x+1}\right)=e^{3x}(x+1)$

    This agrees with what you did, and as stated integration by parts can be used to integrate the RHS:

    $\displaystyle u=x+1\implies dx=dx$

    $\displaystyle dv=e^{3x}\,dx\implies v=\frac{1}{3}e^{3x}$

    Hence:

    $\displaystyle \frac{y}{x+1}=\frac{x+1}{3}e^{3x}-\frac{1}{3}\int e^{3x}\,dx$

    $\displaystyle \frac{y}{x+1}=\frac{x+1}{3}e^{3x}-\frac{1}{9}e^{3x}+C$

    And so:

    $\displaystyle y(x)=\frac{x+1}{9}\left(3(x+1)e^{3x}-e^{3x}+c_2\right)$

    $\displaystyle y(x)=\frac{x+1}{9}\left((3x+2)e^{3x}+c_1\right)$
    Thanks from topsquark and arangu1508
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. RMS Function Integration
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 5th 2015, 10:43 AM
  2. Replies: 0
    Last Post: May 15th 2012, 12:32 AM
  3. [SOLVED] integration of box function
    Posted in the Calculus Forum
    Replies: 6
    Last Post: Jan 25th 2011, 03:36 PM
  4. Function of a Function Quotient Integration
    Posted in the Calculus Forum
    Replies: 5
    Last Post: Jul 29th 2008, 10:13 AM
  5. Function of a function Quotient integration
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Jul 29th 2008, 08:37 AM

/mathhelpforum @mathhelpforum