Riccati differential equation

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)

Answer 1

[Walter Roberson]

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)

Answer 2

[Benjamin]

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..)

[Walter Roberson]

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).