Results 1 to 10 of 10
Like Tree4Thanks
  • 1 Post By princeps
  • 1 Post By princeps
  • 1 Post By princeps
  • 1 Post By princeps

Math Help - Second order ODE reduction to first order solutions

  1. #1
    Newbie
    Joined
    Mar 2012
    From
    UK
    Posts
    23
    Thanks
    1

    Second order ODE reduction to first order solutions

    hey,

    I have this question I have been trying to solve, I got some help and I think I may have solved it but I just wanted to double check here ,

    The question is, Consider the non-linear equation below, use y'=p to reduce it to first order, y(o)=1 and y'(0)=-1

    y''+y(y')3=0

    p=\frac{dy}{dx}= \pm\frac{1}{\sqrt{y^2- 2c}}

    so

     \pm dy \sqrt{y^2- 2c}= dx

    Then

    y'(0)=\pm \frac{1}{\sqrt{{{y}^{2}}(0)-2c}}=\pm \frac{1}{\sqrt{1-2c}}=-1
    \pm \frac{1}{\sqrt{1-2c}}=-1,\,\,\,\pm 1=-\sqrt{1-2c}\,\,squaring\,\,both\,\,sides
    1=1-2c
    c=0
    So\,y'(x)=\pm \frac{1}{y}
    \pm ydy=dx
    \pm \frac{1}{2}{{y}^{2}}=x+c
    \pm \frac{1}{2}{{y}^{2}}(0)=\pm \frac{1}{2}=c
    \therefore y(x)=\sqrt{2x\pm \frac{1}{2}}



    Does this look correct?

    Thanks in advance
    Last edited by Jesssa; April 1st 2012 at 02:38 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2011
    From
    Crna Gora
    Posts
    420
    Thanks
    64

    Re: Second order ODE reduction to first order solutions

    Let y'=v , where v is a function with y as argument ,then :

    y''=\frac{dv}{dy} \cdot v=v'_y \cdot v

    Hence :

    y''+(y')^3=0

    v'_y \cdot v+v^3=0

    v'_y=-v^2

    \int \frac{dv}{v^2}=-\int dy
    Thanks from Jesssa
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Mar 2012
    From
    UK
    Posts
    23
    Thanks
    1

    Re: Second order ODE reduction to first order solutions

    Isn't that the same result?

    you would get

    -1/v = -y

    so y'= 1/y

    only difference is i got +/-, is that the problem in my work having the +/-?

    when I did it I had

    y'' = dp/dy instead of (dp/dy)p
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Nov 2011
    From
    Crna Gora
    Posts
    420
    Thanks
    64

    Re: Second order ODE reduction to first order solutions

    I guess that your " \pm " isn't correct...
    Thanks from Jesssa
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Mar 2012
    From
    UK
    Posts
    23
    Thanks
    1

    Re: Second order ODE reduction to first order solutions

    Ahhh I left out a y!

    What a silly mistake

    The ODE should be

    y''+y(y')3=0


    Sorry,

    p'+yp{{'}^{3}}=0
     \frac{p'}{{{p}^{3}}}=-y
    \frac{dp}{{{p}^{3}}}=-ydy
    -\frac{1}{2}\frac{1}{{{p}^{2}}}=-\frac{1}{2}{{y}^{2}}+c
    {{p}^{2}}=\frac{1}{{{y}^{2}}-2c}
    p=\pm \sqrt{\frac{1}{{{y}^{2}}-2c}}

    Thanks how I got the +/-
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Senior Member
    Joined
    Nov 2011
    From
    Crna Gora
    Posts
    420
    Thanks
    64

    Re: Second order ODE reduction to first order solutions

    I suggest you to use method that I used..It is a standard method for ODEs of this type .
    Thanks from Jesssa
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Mar 2012
    From
    UK
    Posts
    23
    Thanks
    1

    Re: Second order ODE reduction to first order solutions

    So using your method the solution would be

    y(x)=\sqrt{2x + \frac{1}{2}}

    Is that right?

    Edit:

    Hold on that's not right, just working it out quickly now

    I get

    y3 - 3/2 y = 6x - 1/2

    Does that look right?
    Last edited by Jesssa; April 1st 2012 at 03:38 AM.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Senior Member
    Joined
    Nov 2011
    From
    Crna Gora
    Posts
    420
    Thanks
    64

    Re: Second order ODE reduction to first order solutions

    WA , click "show steps"
    Thanks from Jesssa
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Newbie
    Joined
    Mar 2012
    From
    UK
    Posts
    23
    Thanks
    1

    Re: Second order ODE reduction to first order solutions

    ...did not know it could do that haha,

    so its y3 +5/2 y = 6x + 3/2

    thanks!
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Senior Member
    Joined
    Nov 2011
    From
    Crna Gora
    Posts
    420
    Thanks
    64

    Re: Second order ODE reduction to first order solutions

    You could give a name to this thread "Thanksgiving thread "....
    Last edited by princeps; April 1st 2012 at 06:25 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Second-order ODE, reduction of order?
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: March 21st 2012, 07:27 PM
  2. 2nd Order, Homog., Reduction of Order
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: November 27th 2011, 07:36 AM
  3. Reduction of Order
    Posted in the Differential Equations Forum
    Replies: 4
    Last Post: April 24th 2010, 08:21 AM
  4. Replies: 4
    Last Post: August 12th 2008, 05:46 AM
  5. reduction of order?
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 2nd 2006, 12:56 AM

Search Tags


/mathhelpforum @mathhelpforum