Results 1 to 6 of 6
Like Tree4Thanks
  • 2 Post By romsek
  • 1 Post By Prove It
  • 1 Post By HallsofIvy

Thread: 10.09.14t Solve the differential equation

  1. #1
    Super Member bigwave's Avatar
    Joined
    Nov 2009
    From
    Wahiawa, Hawaii
    Posts
    636

    10.09.14t Solve the differential equation

    $\textrm{Solve the differential equation}$
    \begin{align*}\displaystyle
    e^x\frac{dy}{dx}+5e^x y&=4, x>0 \\
    &=
    \end{align*}

    confused about the $\frac{dy}{dx}$
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    5,768
    Thanks
    2417

    Re: 10.09.14t Solve the differential equation

    Quote Originally Posted by bigwave View Post
    $\textrm{Solve the differential equation}$
    \begin{align*}\displaystyle
    e^x\frac{dy}{dx}+5e^x y&=4, x>0 \\
    &=
    \end{align*}

    confused about the $\frac{dy}{dx}$
    one way to go about it is

    $e^x \dfrac{dy}{dx} + 5 e^x y = 4$

    $\dfrac {dy}{dx} + 5 y = 4e^{-x}$

    solving the homogeneous equation

    $\dfrac{dy}{dx} = -5y$

    $y = c_1 e^{-5x}$

    let $y_p = c_2 e^{-x}$

    $-e^{-x}+5 c_2 e^{-x} = 4 e^{-x}$

    $5c_2 - 1 = 4$

    $c_2 = 1$

    $y(x) = c_1 e^{-5x} + e^{-x}$
    Thanks from bigwave and topsquark
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    12,841
    Thanks
    1931

    Re: 10.09.14t Solve the differential equation

    When the equation is first order linear, it's usually expected that the student should use an integrating factor.

    $\displaystyle \begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x} + 5\,y &= 4\,\mathrm{e}^{-x} \end{align*}$

    The integrating factor is $\displaystyle \begin{align*} \mathrm{e}^{\int{5\,\mathrm{d}x}} = \mathrm{e}^{5\,x} \end{align*}$, so multiplying both sides by the integrating factor gives

    $\displaystyle \begin{align*} \mathrm{e}^{5\,x}\,\frac{\mathrm{d}y}{\mathrm{d}x} + 5\,\mathrm{e}^{5\,x}\,y &= \mathrm{e}^{4\,x} \\ \frac{\mathrm{d}}{\mathrm{d}x}\,\left( \mathrm{e}^{5\,x}\,y \right) &= \mathrm{e}^{4\,x} \\ \mathrm{e}^{5\,x}\,y &= \int{ \mathrm{e}^{4\,x} \,\mathrm{d}x} \\ \mathrm{e}^{5\,x}\,y &= \frac{1}{4}\,\mathrm{e}^{4\,x} + C \\ y &= \frac{1}{4}\,\mathrm{e}^{-x} + C\,\mathrm{e}^{-5\,x} \end{align*}$
    Thanks from bigwave
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member bigwave's Avatar
    Joined
    Nov 2009
    From
    Wahiawa, Hawaii
    Posts
    636

    Re: 10.09.14t Solve the differential equation

    that's a different answer?
    Last edited by bigwave; Jun 16th 2017 at 11:40 AM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,336
    Thanks
    2857

    Re: 10.09.14t Solve the differential equation

    Yes, it is.

    The first answer is correct, the second is not. Prove It is correct that the "integrating factor" is e^{5x} but he lost a "4" when he multiplied both sides of the equation by that.
    Thanks from bigwave
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member bigwave's Avatar
    Joined
    Nov 2009
    From
    Wahiawa, Hawaii
    Posts
    636

    Re: 10.09.14t Solve the differential equation

    btw how do we mark our thread as "solved"
    or does it stay open ended
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Differential Equation - how to solve a simple equation..
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: Jun 10th 2012, 03:57 AM
  2. [SOLVED] Solve Differential equation for the original equation
    Posted in the Differential Equations Forum
    Replies: 5
    Last Post: Feb 21st 2011, 01:24 PM
  3. How can I solve this differential equation?
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: Nov 27th 2010, 05:34 AM
  4. Cannot Solve this Differential Equation
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: Feb 17th 2010, 08:19 PM
  5. Replies: 13
    Last Post: May 19th 2008, 08:56 AM

/mathhelpforum @mathhelpforum