This is a homework problem I am trying to solve but I'm not sure if I'm doing it correctly, because it seems deceptively simple.

Let $\alpha$ be a non-negative real constant. The differential equation

$$ x^2y\prime\prime +xy\prime+(x^2-\alpha ^2)y=0 $$

has solutions called Bessel functions. Without solving the differential equation, compute the Wronskian of two Bessel functions by using Abel's Theorem.

Here is what I did

$$ y\prime\prime + \frac{1}{x}y\prime + \frac{(x^2-\alpha^2)}{x^2}y = 0 \\ W(y_1,y_2) = c\cdot \exp\left[-\int p(t)dt\right] \\ = c \cdot \exp\left[-\int \frac{1}{x} dt\right] \\ =c \cdot \exp[-\ln|x|] \\ =cx^{-1} = c\cdot\frac{1}{x} $$

Is this correct or am I missing something? Any help would be greatly appreciated.

  • $\begingroup$ Do you know what $c$ it is? $\endgroup$ – Mhenni Benghorbal Feb 27 '14 at 5:19
  • $\begingroup$ Well $c$ is just a constant that would be determined from initial conditions. $\endgroup$ – RXY15 Feb 27 '14 at 5:25
  • $\begingroup$ Your solution is correct. $\endgroup$ – Mhenni Benghorbal Feb 27 '14 at 5:33

Coincidentally , I have the exact same homework and was thinking the same thing after getting it done just like this. Using the Abel's identity, your solution couldn't be more correct.

  • $\begingroup$ Welcome to Mathematics Stack Exchange! Unfortunately, this does not provide an answer to the question. $\endgroup$ – Glorfindel Mar 26 '17 at 17:06

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