Riccati differential equation

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Benjamin on 25 May 2011
Hello, i've got a riccati differential equation and matlab (dsolve) is not able to solve it. has anyone an idea how i can get the result?
The equation is:
dy/dx = 1/(2x^2y-1)

Answers (2)

Walter Roberson
Walter Roberson on 25 May 2011
To confirm, is your equation
diff(y(x), x) = 1/(2*x^2*y(x)-1)
If so, then Maple gives this ugly solution:
C1+(I*AiryBi(-2*y(x)/(2*I)^(2/3))/((2*I)^(1/3)*x)-AiryBi(1, -2*y(x)/(2*I)^(2/3)))/(I*AiryAi(-2*y(x)/(2*I)^(2/3))/((2*I)^(1/3)*x)-AiryAi(1, -2*y(x)/(2*I)^(2/3))) = 0
where C1 is the arbitrary constant of integration and "I" is sqrt(-1)

Benjamin on 25 May 2011
yes, this is the equation, and i think the solution could be right... is there any possibility to get it with matlab? (or more exactly: i need the values of the n^th derivation - n-> infinity - at x = 0 (i still have got the constant C1, so thats not the problem) for the further calculations... maybe there is an easier way to calculate... please give me a hint if it is so..)
  1 Comment
Walter Roberson
Walter Roberson on 25 May 2011
I do not have any insight to efficient ways to calculate the derivatives of the above.
I do not have the Symbolic Toolbox myself (I have Maple), so I do not know what happens if you
evalin(symengine, 'dsolve(diff(y(x),x) = 1/(2*x^2*y(x)-1))')
The Airy* solution can be converted to either BesselI or hypergeom and worked with from that point, but I see that MuPad does have airyAi and airyBi functions, so it appears differentiating from the Airy* solution should work (if perhaps slowly).

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