Results 1 to 6 of 6

Math Help - *Urgent* Differential Equations

  1. #1
    Newbie
    Joined
    Sep 2008
    From
    Halifax, Nova Scotia
    Posts
    19

    *Urgent* Differential Equations

    Need to find the general solution of:
    y" + (1+2i)y' + (i-1)y = 0
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Nov 2008
    Posts
    82
    Quote Originally Posted by gidget View Post
    Need to find the general solution of:
    y" + (1+2i)y' + (i-1)y = 0
    There are numerous methods to solve 2nd Order Linear DE's, has your teacher specified a method to use? i.e. Method of Un-Determined Coefficients, Linear D operators (Forming an Auxillary eqn etc) ?

    If none is specified, for simplicity I would recommend the Linear D operator technique,

    If you let me know what method your meant to use, I will post a solution using that method,

    Regards,

    David
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2008
    From
    Halifax, Nova Scotia
    Posts
    19
    yeah he uses linear D operators I just have no clue how to attempt this or where to go with this problem.
    thank you!!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Banned
    Joined
    Nov 2008
    Posts
    82
    Okay, just for complete clarrification, in your notes does he convert the above D into the form

    (D+ a)(D + b)y = 0 to solve ??

    and then solve from here using the properties of the linear D operator or does your teacher then further impose that y must take the form Ae^mx ? and solve that way?

    Sorry, just want to get it 100% clear before I go ahead with the solution,

    Regards,

    David
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Sep 2008
    From
    Halifax, Nova Scotia
    Posts
    19
    D ^2 + (1+2i)D + i -1 =0

    This is what was also given to us.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Banned
    Joined
    Nov 2008
    Posts
    82
    ahh okay cool

    Basically what your teacher is saying to do is to form what is known as the auxillary eqn,

    here you have a linear DE of the form

    a* D^2y + b*Dy + cy = 0
    or

    (a*D^2 + b*D + c)y = 0

    let y = Ae^(mx)

    Subsitutuion yeilds,

    Ae^(mx)*(a*m^2 + b*m + c) = 0

    As Ae^(mx)~= 0 unless A = 0 (which of course would be the trivial solution) we are arrive at the Auxillary equation,

    a*m^2 + b*m + c = 0

    here a = 1; b = (1 + 2i); c = i - 1

    obviously this is a mere quadratic and thus form the quadratic formula is used to solve the Auxillary eqn,

    m = (- (1+2i) +- sqrt( (1+2i)^2 - 4(1)(i-1) ))/(2(1))
    note (1 + 2i)^2 - 4(i-1) = 4i^2 + 4i + 1 - 4i + 4 = 1
    thus,
    m = (-(1 + 2i) +- 1)/2
    Hence
    m1 = i , m2 = -(1+i)

    as y = e^(mx) and by employing the superposition principle, the solution takes the form,

    y = Ae^(m1*x) + Be^(m2*x); where A, B are any complex values
    = Ae^(ix) + Be^(-(1+i)x)

    And viola your done!

    Hope this helps,

    Regards,

    David

    note - you could put the expression of e^aix into terms of sin(ax), cos(ax) using Eulers Identity - but once again it all depends on what your teacher wants, the solution provided is perfectly correct
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. differential equations homework urgent!
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: December 23rd 2008, 03:33 PM
  2. differential equations urgent homework!
    Posted in the Differential Equations Forum
    Replies: 4
    Last Post: December 23rd 2008, 02:48 AM
  3. Ordinary Differential Equations!Urgent Help Neede
    Posted in the Differential Equations Forum
    Replies: 20
    Last Post: June 14th 2008, 09:13 PM
  4. urgent help needed with differential equations
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 2nd 2007, 04:49 AM
  5. <help> Differential Equations... MAPS (urgent)
    Posted in the Calculus Forum
    Replies: 3
    Last Post: August 11th 2007, 04:44 AM

Search Tags


/mathhelpforum @mathhelpforum