Finding Bessel Solutions for a Differential Equation with a Transformed Format

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Homework Statement


u''-bc (x^m) u =0

Homework Equations


How can I write the general solution in terms of Bessel function?

The Attempt at a Solution



This form is a transformed vresion of y'+by^2=cx^m with dummy variable by=1/u *du/dx
 
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ramtin said:

Homework Statement


u''-bc (x^m) u =0


Homework Equations


How can I write the general solution in terms of Bessel function?


The Attempt at a Solution



This form is a transformed vresion of y'+by=cx^m with dummy variable by=1/u *du/dx

I'm curious why you would go to the trouble of transforming your original DE when you can solve it directly. It's just a constant coefficient DE with a non-homogeneous polynomial term.
 
LCKurtz said:
I'm curious why you would go to the trouble of transforming your original DE when you can solve it directly. It's just a constant coefficient DE with a non-homogeneous polynomial term.

it was y'+by^2=cx^m ,Iforgot to type the y squared power
 
Here is what Maple gives, for what it's worth:

bessel.jpg
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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