Results 1 to 7 of 7

Math Help - Differentiate Solution and Match Up with Equation

  1. #1
    s3a
    s3a is offline
    Super Member
    Joined
    Nov 2008
    Posts
    597

    Differentiate Solution and Match Up with Equation

    For C, can someone show me step-by-step how to isolate the function y and/or its derivatives? I said "and/or" just in case, I cannot isolate the function y itself but can for its derivatives so that I can say that the equation C is the general solution of one the numbers.

    Any help would be greatly appreciated!
    Thanks in advance!
    Attached Thumbnails Attached Thumbnails Differentiate Solution and Match Up with Equation-output.jpeg  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2

    Re: Differentiate Solution and Match Up with Equation

    Use implicit differentiation and the product rule. What do you get as a first step?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,315
    Thanks
    1223

    Re: Differentiate Solution and Match Up with Equation

    Quote Originally Posted by s3a View Post
    For C, can someone show me step-by-step how to isolate the function y and/or its derivatives? I said "and/or" just in case, I cannot isolate the function y itself but can for its derivatives so that I can say that the equation C is the general solution of one the numbers.

    Any help would be greatly appreciated!
    Thanks in advance!
    1. is a standard second order linear constant coefficient ODE. How do you solve those?

    2. Rewrite it as

    \displaystyle \begin{align*} \frac{dy}{dx} &= -\frac{2xy}{x^2 - 6y^2} \\ \frac{dx}{dy} &= \frac{6y^2 - x^2}{2xy} \\ \frac{dx}{dy} &= \frac{6y^2}{2xy} - \frac{x^2}{2xy} \\ \frac{dx}{dy} &= 3\left(\frac{y}{x}\right) - \frac{1}{2}\left(\frac{x}{y}\right) \end{align*}

    and then make the substitution \displaystyle v = \frac{x}{y}...

    3. Another second order linear constant coefficient ODE...

    4. The DE is separable.

    5. The answer will be the whichever one is left over...
    Follow Math Help Forum on Facebook and Google+

  4. #4
    s3a
    s3a is offline
    Super Member
    Joined
    Nov 2008
    Posts
    597

    Re: Differentiate Solution and Match Up with Equation

    I'm attaching the pdf which contains that segment of my work but I am now stuck at the next step. I also don't see how it's useful in differential equations to have solutions with y not isolated even though the derivatives y' (and so on) are isolated or basically if at least one of the equations has a problem with the isolation.
    Attached Files Attached Files
    Follow Math Help Forum on Facebook and Google+

  5. #5
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2

    Re: Differentiate Solution and Match Up with Equation

    So, divide top and bottom of the RHS of your final result by -3. Does that start to look more familiar?

    I also don't see how it's useful in differential equations to have solutions with y not isolated even though the derivatives y' (and so on) are isolated or basically if at least one of the equations has a problem with the isolation.
    You can still obtain useful information about the solutions, even if they're defined implicitly. For example, computers can graph implicit solutions.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    s3a
    s3a is offline
    Super Member
    Joined
    Nov 2008
    Posts
    597

    Re: Differentiate Solution and Match Up with Equation

    Ackbeet: I noticed that C matches #2 when I divide top and bottom by -3. Does this mean that the antiderivative of #2 is the general solution for differential equation #2?

    Prove It: You mentioned a few stuff that I did not cover so I'll look at it when I learn the new stuff to get this question from a different perspective so thank you still.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2

    Re: Differentiate Solution and Match Up with Equation

    Quote Originally Posted by s3a View Post
    Ackbeet: I noticed that C matches #2 when I divide top and bottom by -3. Does this mean that the antiderivative of #2 is the general solution for differential equation #2?
    I wouldn't quite phrase it that way. You can't integrate # 2 directly. Say, rather, that Equation C is the general solution of DE # 2. On the other hand, the process of solving differential equations is essentially integrating. So maybe you can say that the integral of DE # 2 is Equation C.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Differentiate an equation
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 17th 2011, 06:22 PM
  2. Replies: 1
    Last Post: October 17th 2011, 05:19 AM
  3. Replies: 1
    Last Post: February 13th 2010, 09:25 AM
  4. Differentiate the equation
    Posted in the Calculus Forum
    Replies: 3
    Last Post: August 27th 2009, 05:30 PM
  5. How do you Differentiate this equation?
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 23rd 2009, 03:42 PM

Search Tags


/mathhelpforum @mathhelpforum