Results 1 to 2 of 2

Math Help - Finding the General Solution of a 2nd Order Linear Ordinary Differential Equation

  1. #1
    Member
    Joined
    Nov 2009
    Posts
    92

    Finding the General Solution of a 2nd Order Linear Ordinary Differential Equation

    Question: Find the general solution to the equation:  t^2y'' - 2y = t .

    Attempt at solution: The main catch to this question that causes me problems is the fact that the coefficients are not numbers but rather variables such as t in the question. I know to use the method of undetermined coefficients to solve this question, but i am just stuck on that part!

    Thanks to all who contribute!
    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
    The equation is of the Cauchy Euler type.

    Make the substitution

    z=\ln(t) then

    \frac{dy}{dt}=\frac{dy}{dz}\frac{dz}{dt}=\frac{1}{  t}\frac{dy}{dz}

    y''=\frac{d}{dt}\frac{dy}{dt}=-\frac{1}{t^2}\frac{dy}{dz}+\frac{1}{t^2}\frac{dy^2  }{dz^2}

    Can you finish from here?

    This gives the ODE

    t^2(-\frac{1}{t^2}\frac{dy}{dz}+\frac{1}{t^2}\frac{dy^2  }{dz^2})-2y=e^{z}

    y''-y'-2y=e^{z}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] general Solution to a Linear, Homogenious Differential Equation
    Posted in the Differential Equations Forum
    Replies: 13
    Last Post: September 16th 2010, 05:40 PM
  2. help Second-order linear ordinary differential equation
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: March 8th 2010, 06:08 AM
  3. Help with Third-order linear ordinary differential equation
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: December 4th 2009, 02:34 PM
  4. general solution to linear first order differential equations
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: February 4th 2009, 09:58 PM
  5. Replies: 11
    Last Post: May 30th 2008, 02:07 PM

Search Tags


/mathhelpforum @mathhelpforum