Results 1 to 4 of 4

Math Help - Clairaut's/Riccati's Equation Help Please!

  1. #1
    Newbie
    Joined
    Feb 2014
    From
    United States
    Posts
    2

    Clairaut's/Riccati's Equation Help Please!

    Hello all, this is my first time posting, and I need some help, big time!
    Alright so my teacher gave us a homework assignment without much in the way of hints for how to approach the problems, so I would like to post the 3 problems here. The first two I have an answer for but would like verification that they are correct, and the last one I have no idea of how to approach.

    1. Use Clairaut's Equation to show that the equation:
    y = xy' + (y')2
    Has a singular solution that is the envelope of the one parameter family of solutions
    Note: I am pretty confident about this one, so if you don't feel like spending the time with this one, I'd totally understand
    My answer was:
    General solution- y = c2+cx
    Particular Solution - y = -x2/4
    Because it involves a graph, I'll attach an image of the work.

    2. Solve the equation dy/dx = y2 - 2/x2
    if it is known that y1 = 1/x is a particular solution of the equation. Hint: make y = z + 1/x

    Okay so I used that substitution to transform the equation into a Bernoulli equation, and after a bit of arithmetic I got y = -3x2/(x3+c) + 1/x
    The problem is that I do not understand where the y1 = 1/x comes into play. He wrote it just like that, with the 1 as a subscript (as opposed to y(1) = 1/x), so I just have no idea what to do with it.
    I will again attach a picture in case you would like to follow my logic

    3. Find the condition for which the equation:
    (x-y)dx + (y+x)dy = 0

    has an integrating factor of the form (lambda = u):
    u = u(x2 + y2)
    also, find the integrating factor and solve the equation.
    Okay... so I understand the concept of integrating factors and exact equations, but this just seems bizarre to me, what's with the two lambdas? Can anyone offer any guidance on this one?
    I'll attach a picture of the test so that you could see exactly how it was written.

    Any help that can be provided would be greatly appreciated, I'm getting a little desperate!

    Thank you in advance!
    Note: I had to zip all the pictures together because they were too large of a file.

    Pictures.zip
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9

    Re: Clairaut's/Riccati's Equation Help Please!

    Do not solve, (x-y) + (y+x)y' = 0, via integrating factor. It would not work.

    Instead solve for y,

    y' = (y-x)/(y+x)

    So the equation is of the form,
    y' = h(x,y)
    Where h(x,y) is a homosexual function i.e. h(ax,ay) = a*h(x,y).
    Thus, use the substitution z = y/x to solve this equation.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,333
    Thanks
    894

    Re: Clairaut's/Riccati's Equation Help Please!

    Quote Originally Posted by ThePerfectHacker View Post
    Where h(x,y) is a homosexual function i.e. h(ax,ay) = a*h(x,y).
    a what??

    (I think you mean homogeneous)
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Feb 2014
    From
    United States
    Posts
    2

    Re: Clairaut's/Riccati's Equation Help Please!

    The problem is that we need to solve it by integrating factor. Also, our professor told us that the integrating factor he listed there means that it has to be in terms of x^2 + y^2. Can anyone help with this?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Clairaut's equation.
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: October 23rd 2013, 01:52 AM
  2. Riccati's Equation
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: November 20th 2011, 07:00 AM
  3. Clairaut's Equation
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: June 27th 2010, 06:15 PM
  4. Riccati Equation
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: April 3rd 2009, 12:19 PM
  5. help with riccati equation
    Posted in the Calculus Forum
    Replies: 3
    Last Post: February 17th 2009, 04:00 PM

Search Tags


/mathhelpforum @mathhelpforum