Results 1 to 2 of 2

Math Help - nonhomogenous equations

  1. #1
    Junior Member
    Joined
    Oct 2008
    Posts
    70

    nonhomogenous equations

    Consider the differential equation ay''+by'+cy=d where d is a constant. Show that every solution of the equation approaches d/c as t goes to infinity. What happens if c=0? What if b=0?

    Setting Y(t)=A, and following from that Y'(t)=0 and Y''(t)=0. I plugged these values in to the differential equation and got that A=d/c but I confused about how to find the homogenous solution and how do we know that that will go to 0 as t goes to infinity.

    thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by morganfor View Post
    Consider the differential equation ay''+by'+cy=d where d is a constant. Show that every solution of the equation approaches d/c as t goes to infinity. What happens if c=0? What if b=0?

    Setting Y(t)=A, and following from that Y'(t)=0 and Y''(t)=0. I plugged these values in to the differential equation and got that A=d/c but I confused about how to find the homogenous solution and how do we know that that will go to 0 as t goes to infinity.

    thanks!
    I don't think it is true unless you restrict at lease some of the values of a,b and c.


    To solve the complimentry solution use the ansatz y=e^{rt} \implies y'=re^{rt} \implies y''=r^2e^{rt}

    Then plut into the homogenous equation to get

    ay''+by'+cy=0 \iff ar^2e^{rt}+bre^{rt}+ce^{rt}=0

    factoring gives

    e^{rt}(ar^2+br+c)=0

    Since the exponential function is never zero the quadratic must be zero.
    Using the quadratic equation we gt

    r=\frac{-b\pm \sqrt{b^2-4ac}}{2a}

    So the complimetery solution to the equation is

    y_c=c_1e^{ \frac{-b + \sqrt{b^2-4ac}}{2a}t}+c_2e^{\frac{-b- \sqrt{b^2-4ac}}{2a}t}

    This will not always tend to zero as t goes to infinity. I.e if b,c > 0 \text{ and } a < 0

    The solution of the homoenous problem goes to infinity.

    I hope this helps
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Solving a nonhomogenous D.E.
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: February 3rd 2010, 05:15 AM
  2. solving a nonhomogenous recursion
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: May 23rd 2009, 09:03 AM
  3. Nonhomogenous Linear Equation
    Posted in the Differential Equations Forum
    Replies: 4
    Last Post: February 27th 2009, 03:24 PM
  4. Replies: 4
    Last Post: December 27th 2008, 08:56 AM
  5. second order linear nonhomogenous
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 7th 2007, 05:06 PM

Search Tags


/mathhelpforum @mathhelpforum