Results 1 to 14 of 14

Math Help - solve this Differential equation by indicial equation

  1. #1
    Senior Member
    Joined
    Feb 2008
    Posts
    321

    solve this Differential equation by indicial equation

    Attached Thumbnails Attached Thumbnails solve this Differential equation by indicial equation-5566.jpg  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by szpengchao View Post
    Given the form of the DE, I would assume that the first part requires you to find a solution of the form Ax^n.

    Where are you stuck in the second part?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Feb 2008
    Posts
    321

    problem

    i have problem with the first part
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by szpengchao View Post
    Hint:
    z=\ln(x)

    Use the chain rule

    \frac{dy}{dx}=\frac{dy}{dz}\frac{dz}{dx}=\frac{1}{  x}\frac{dy}{dz}

    Now we need the 2nd derivative using the product and chain rule

    \frac{d^2y}{dx^2}=-\frac{1}{x^2}\frac{dy}{dz}+\frac{1}{x}\left( \frac{d^2y}{dz^2} \cdot \frac{1}{x}\right)

    from here sub into the equation

    Good luck
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member
    Joined
    Feb 2008
    Posts
    321

    no

    no...i mean the 1st question...not the second
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by szpengchao View Post
    no...i mean the 1st question...not the second
    Are you talking about using the method of frobenius to obtain series solutions to the ODE?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Senior Member
    Joined
    Feb 2008
    Posts
    321

    yeah

    yeah.... i have problem with choosing point for this particular question...
    would you like to post ur steps?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Senior Member
    Joined
    Feb 2008
    Posts
    321

    choosing ordinary point

    well, x=0 is a regular singular point, right?
    then , i set y= sigma( A_n * x^(n+b)) into the equation. then i got something like:

    sigma( A_n * (n^2+1) = 0) well, this means A_n = 0 for all n...then how can i continue?
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by szpengchao View Post
    well, x=0 is a regular singular point, right?
    then , i set y= sigma( A_n * x^(n+b)) into the equation. then i got something like:

    sigma( A_n * (n^2+1) = 0) well, this means A_n = 0 for all n...then how can i continue?
    y=\sum_{n=0}^{\infty}c_nx^{n+r}

    y'=\sum_{n=0}^{\infty}c_n(n+r)x^{n+r-1}


    y''=\sum_{n=2}^{\infty}c_n(n+r)(n+r-1)x^{n+r-2}

    x^2y''+xy'+y=0

    x^2\sum_{n=0}^{\infty}c_n(n+r)(n+r-1)x^{n+r-2}+x\sum_{n=0}^{\infty}c_n(n+r)x^{n+r-1}+ \sum_{n=0}^{\infty}c_nx^{n+r}=0

    \sum_{n=0}^{\infty}c_n(n+r)(n+r-1)x^{n+r}+\sum_{n=0}^{\infty}c_n(n+r)x^{n+r}+ \sum_{n=0}^{\infty}c_nx^{n+r}=0

     c_nx^{n+r}\left(\sum_{n=0}^{\infty}(n+r)(n+r-1)+(n+r)+ 1 \right)=0

     c_nx^{n+r}\left(\sum_{n=0}^{\infty}(n+r)[(n+r-1)+1]+ 1 \right)=0

     c_nx^{n+r}\left(\sum_{n=0}^{\infty}(n+r)^2+ 1 \right)=0

    I think this is where you are stuck.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Senior Member
    Joined
    Feb 2008
    Posts
    321

    yup

    yup...then? how can that be zero?
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by szpengchao View Post
    yup...then? how can that be zero?
    I don't know besides the trivial solution. Maybe someone can find my mistake.

    Follow Math Help Forum on Facebook and Google+

  12. #12
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by TheEmptySet View Post
    I don't know besides the trivial solution. Maybe someone can find my mistake.

    Using the substitution given in the second part, I get y = A \sin (\ln x) + B \cos (\ln x) ......
    Last edited by mr fantastic; May 19th 2008 at 01:22 AM. Reason: Deleted last sentence
    Follow Math Help Forum on Facebook and Google+

  13. #13
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by mr fantastic View Post
    Given the form of the DE, I would assume that the first part requires you to find a solution of the form Ax^n.

    Where are you stuck in the second part?
    Do this and you get n^2 + 1 = 0 \Rightarrow n = \pm i.

    Therefore

    y = A_1 x^i + A_2 x^{-i} = A_1 e^{i \ln x} + A_2 e^{-i \ln x} = B_1 \cos (\ln x) + B_2 \sin (\ln x)

    by the magic of Euler's equation and the usual identity.
    Follow Math Help Forum on Facebook and Google+

  14. #14
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Thanks Mr. Fantastic,

    I forgot that c_0 \ne 0 in this method so

    c_nx^{n+r}\left(\sum_{n=0}^{\infty}(n+r)^2+ 1 \right)=0 \iff c_0x^{r}(r^2+1)+c_nx^{n+r}\left(\sum_{n=1}^{\infty  }(n+r)^2+ 1 \right)=0

    so c_n=0, \forall n \ge 1

    r^2=1 \iff r=\pm i

    Then the rest follows.

    Thanks again!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Solve Differential equation for the original equation
    Posted in the Differential Equations Forum
    Replies: 5
    Last Post: February 21st 2011, 01:24 PM
  2. How can I solve this differential equation?
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: November 27th 2010, 05:34 AM
  3. Confusing indicial equation
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: May 27th 2010, 02:28 AM
  4. Help solve a differential equation
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 26th 2007, 01:19 AM
  5. indicial equation
    Posted in the Algebra Forum
    Replies: 5
    Last Post: April 29th 2006, 10:27 PM

Search Tags


/mathhelpforum @mathhelpforum