Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Riccati Method

  1. Aug 25, 2005 #1
    I want to read a bit more example on using the Riccati Method when solvng D.E, who can help me?
  2. jcsd
  3. Aug 25, 2005 #2


    User Avatar
    Science Advisor
    Homework Helper

  4. Aug 25, 2005 #3
    Thanks very much~
    But how can I get one of the soultion of
    y'= [2cos^2(x) - sin^2(x) + y^2]/ 2cos(x) =0 ? (the second example in the reading)
    If can't~ no use of the Riccati method~
  5. Aug 26, 2005 #4


    User Avatar
    Science Advisor
    Homework Helper

    What's the problem? For one, they give the solution right on the page, so I don't know why you think you can't do it (unless you mean that you're trying to solve it yourself and haven't looked at the answer). Second of all:

    [tex]\frac{2\cos ^2(x) - \sin ^2(x) + y^2}{2\cos (x)} = \left (\frac{2\cos ^2(x) - \sin ^2(x)}{2\cos (x)}\right )y^0 + (0)y^1 + \frac{1}{2\cos (x)}y^2[/tex]

    Also, I don't know why you're setting it to 0, you don't need to solve y' = 0. You have an equation for y' in terms of f and x, and you make a substitution for z to get you a linear equation which you can solve. You're given that y1 = sin(x) is a solution, so set:

    [tex]z = \frac{1}{y - y_1}[/tex]

    If you isolate y in that equation, then you can express y in terms of z and y1, and can even express y' in terms of z and y1. You find those expressions and substitute them into your Ricatti equation. You then perform the algebraic manipulations to isolate z', and on the right side you should end up with a linear expression. Solve for z easily, then substitute back to find y.
  6. Aug 26, 2005 #5
    What you means is unless one of the soultion is given i.e y1 otherwise we cant solve this D.E?
  7. Aug 26, 2005 #6


    User Avatar
    Science Advisor
    Homework Helper

    I don't know if you even read the link I gave you, since if you did there should be no question as to how to solve it. You also missed the seventh sentence on that page:

    "Without knowing at least one solution, there is absolutely no chance to find any solutions to such an equation."
  8. Aug 27, 2005 #7


    User Avatar
    Science Advisor
    Homework Helper

    You know AKG, I really think that statement should be qualified: It depends of course on what P(x), Q(x), and R(x) are. By use of the transformation:


    The Ricccati equation is converted to a linear second-order ODE:


    In some cases, this equation can be solved directly or via power series.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Riccati Method
  1. Special riccati equation (Replies: 13)

  2. Riccati's ODE variant. (Replies: 5)