Results 1 to 3 of 3

Math Help - linear differential equations

  1. #1
    Junior Member
    Joined
    Mar 2008
    From
    Perth
    Posts
    26

    linear differential equations

    show that the differential eqn x^2 * dy/dx -y = 2x^3*e^(-1/x)
    is linear, and solve it using the integrating factor method.

    I have no idea how to even get started on this problem, if someone could show me in a few steps it would really help me out. I have alot more of these to do.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by samdmansam View Post
    show that the differential eqn x^2 * dy/dx -y = 2x^3*e^(-1/x)
    is linear, and solve it using the integrating factor method.

    I have no idea how to even get started on this problem, if someone could show me in a few steps it would really help me out. I have alot more of these to do.
    y' - \frac{1}{x^2}y = 2xe^{-1/x}, let p(x) = e^{\int -\frac{1}{x^2} dx}.

    The solution is given by y = \frac{1}{p(x)} \left\{ \int 2xp(x)e^{-1/x} dx  + C \right\}.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member PaulRS's Avatar
    Joined
    Oct 2007
    Posts
    571
    Ok, suppose you have a differential equation of the form: f'(x)+a(x)\cdot{f(x)}=b(x)

    Let f(x)\cdot{e^{\int_c^xa(t)dt}}=q(x) (1) be an auxiliary function then: f'(x)\cdot{e^{\int_c^xa(t)dt}}+a(x)\cdot{f(x)}\cdo  t{e^{\int_c^xa(t)dt}}=q'(x)

    But note that: \left(f'(x)+a(x)\cdot{f(x)}\right)\cdot{e^{\int_c^  xa(t)dt}}=b(x)\cdot{e^{\int_c^xa(t)dt}}=q'(x)

    Thus, by the FTC: \int_c^xb(z)\cdot{e^{\int_c^za(t)dt}}dz=q(x)-q(c)=q(x)-f(c) check in (1) that q(c)=f(c)

    Thus we have by (1) that: e^{-\int_c^xa(t)dt}\cdot{\int_c^xb(z)\cdot{e^{\int_c^z  a(t)dt}}dz}+f(c)\cdot{e^{-\int_c^xa(t)dt}}=f(x)

    This is the general solution to the first order equation

    So suppose you are given f(0) for example, then you should let c=0 above
    Last edited by PaulRS; June 4th 2008 at 06:03 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Linear differential equations.
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: September 10th 2010, 03:20 AM
  2. Linear Differential Equations
    Posted in the Algebra Forum
    Replies: 1
    Last Post: April 26th 2009, 01:43 PM
  3. Replies: 1
    Last Post: May 15th 2008, 09:23 PM
  4. linear differential equations??
    Posted in the Calculus Forum
    Replies: 5
    Last Post: April 20th 2008, 10:44 AM
  5. Replies: 5
    Last Post: July 16th 2007, 05:55 AM

Search Tags


/mathhelpforum @mathhelpforum