-1
$\begingroup$

Can you help me find the leading asymptotic behaviors about the irregular singular point $x=0$ of $$x^4 \frac{d^2y}{dx^2}+ \frac{1}{4}y=0$$

So far I have got $y(x) = c_{1}8\exp(2/x)+c_{2}8\exp(-2/x)$, is this on the right track for the answer?

  • 1
    What do you mean by asymptotic behavior?2012-02-23
  • 0
    I it means is the equation analytic. Im not sure tho.2012-02-23
  • 0
    Mathematica gives the following: inputting "DSolve[x^4 f''[x] + 1/4 f[x] == 0, f[x], x]" gives a general solution of the form "E^(I/(2 x)) x C[1] - I E^(-(I/(2 x))) x C[2]", for some constants C[1] and C[2]. Thus, it seems likely that some variable substitution of your original equation leads a differential equation with constant coefficients. Note that your equation is *irregular singular* at $x=0$, which explains why the solution has an essential singularity around this point.2012-02-23
  • 0
    I have got $y(x) ~ c_{1}8\exp{2/x}+c_{2}8\exp{-2/x}$, is this on the right track for the answer?2012-02-23

1 Answers 1