Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.