# How can you find the Wronskian of Bessel Functions?

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.

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