Need help to solve a diff. equation

[mathphys]

i need help to solve this equation? is there anybody has some comments..

y''+ (1/x)y'+{[cx^4-(ax^2-b)]/[(x^2)*(ax^2+b)]}y=0

[HallsofIvy]

Ouch! That's a linear differential equation, which is good, but with nastily varying coefficients. Probably the best way to do this is to try to solve for y as a Taylor's series (or, more correctly, approximate y by a Taylor polynomial).

Any Taylor Polynomial can be written as
y= y(0)+ y'(0)x+ y"/2 x^2+ ...+ y(n)(0) x^n/n! where "y(n)(0)" is the nth derivative at 0. You are not given initial conditions (which would make this much simpler) so assume that y(0)= C1 and y'(0)= C2. Now you can use that differential equation to find y"(0) in terms of C1 and C2.

Differentiating both sides of the equation (which will be messy) will give you a formula for the third derivative of y in terms of y, y' and y". You can use that to find the third derivative of y at 0 in terms of C1 and C2.

Differentiate AGAIN to get a formula for the fourth derivate and use the values of y, y', y" and y"' that you have calculated to find y""(0) in terms of C1 and C2.

Continue that as long as you can stand (it will get messier and messier with each differentiation) to find as many coefficients of the Taylor's polynomial as you can.

[Nelykur]

thank you Arhat for your clarification but but, I dont understand the structure and meaning of this sentence She has borne him 6 sons. can you make the idea more clear please and thank you again