I'm working on what should be a relatively straightforward differential equation. The problem says that the Laplace transform of Bessel's equation leads to (s^2 +1)f'(x) +sf(s)=0. And asks to solve for f(s). I don't understand what is the dependent variable of the function, and if the other letter represents a constant or a variable. I've looked up Laplace transforms of Bessel's equation, but do not understand the treatment given, and cannot find this exact form of the transform written anywhere. My question is what do the letters x and s represent?
[Math] Laplace transform of Bessel’s equation
bessel functionslaplace transformordinary differential equations
Best Answer
I think there is a typo, see this book and we have:
$$(s^2 +1)f'(s) +sf(s)=0$$
This leads to:
$$\displaystyle \int \dfrac{df}{f} = - \int \dfrac{s~ ds}{s^2+1}$$
After integrating, you arrive at:
$$f(s) = \dfrac{c}{\sqrt{s^2+1}}$$
As an aside, this problem comes from the Bessel equation:
$$x^2 y''(x) + x y'(x) + x^2 y(x) = 0$$