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I'm stuck with a differential equation

  1. Apr 1, 2010 #1
    Please Help ................... Urgent

    Hello ppl
    I stuck with a differential equation and all I need is your kind help. While solving a problem I got one eqaution as given below


    ((Ax+B)^4)y''''+ ((Ax+B)^2)y''+(i/const.)y=0


    here A, B, and const are constants

    i denotes complex no (iota).

    I need your kind help. Please suggest me the possible way to solve this equation.

    Thanks in advance
     
  2. jcsd
  3. Apr 1, 2010 #2
    Re: Please Help ................... Urgent

    First, change variable

    x = (tA-B)/A,

    where t is new variable. Your ODE in new variable is

    (At)^4y''''+ (At)^2y''+(i/const.)y=0 (2)

    this is an "equipotential" equation, also known as an "Euler-type equation".

    Then seek solution in form y(t)=t^z. Substitution y(t)=t^z to (2) leads to

    t^z(A^4 z^4-6 A^4 z^3+11 A^4 z^2-6 A^4 z+A^2 z^2-A^2 z+(i/const.))=0,

    There are four solutuons of algebraic equation (z1,z2,z3,z4)

    A^4 z^4-6 A^4 z^3+11 A^4 z^2-6 A^4 z+A^2 z^2-A^2 z+(i/const.)=0,

    so the general solution of ode (2) is as follows

    y(t)=C1t^z1+...+C4t^z4

    where C1...C4 are arbitrary constants.
     
  4. Apr 1, 2010 #3
    Re: Please Help ................... Urgent

    Thanks kosovtsov for your kind help.
     
  5. Apr 1, 2010 #4
    Re: Please Help ................... Urgent

    Sir can you briefly explain about these "Euler Type Equations". Can you give me some references for these equations.
     
  6. Apr 2, 2010 #5
  7. Apr 2, 2010 #6
    Re: Please Help ................... Urgent

    Sir
    Can we treat i/const. as a constant. I am confused about it. Please help me in this regard. Can you suggest me some book having similar kind of problem. I will be thankful to you.
     
  8. Apr 2, 2010 #7
    Re: Please Help ................... Urgent

    Yes, I have considered i/const as a constant. But real meaning of it belong exclusively to the physical problem setting. I know nothing on the physical base of your problem.
     
  9. Apr 2, 2010 #8
    Re: Please Help ................... Urgent

    Sir, The physical problem belongs to planer wave where the physical medium which carries this wave have tapered cross section. Can this thing help you or I have to explain the problem in detail. All I want to ask (i/const.) is a complex no. so how can we take it as real no.
     
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