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Riccati's equation and Bessel functions
- Thread starter Fernando Revilla
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Re: Riccati's equation an Bessel functions
This is the answer I have posted there:
This is the answer I have posted there:
Does anyone know an alternative?
chisigma
Well-known member
Re: Riccati's equation an Bessel functions
The non linear first order Riccati ODE...
$$ y^{\ '} = x^{2} + y^{2}\ (1)$$
... can be transformed into a linear second order ODE with the substitution...
$$y = - \frac{u^{\ '}}{u} \implies y^{\ '} = - \frac{u^{\ ''}}{u} + (\frac{u^{\ '}}{u})^{2}\ (2)$$
... so that we have to engage the ODE...
$$u^{\ ''} + x^{2}\ u =0\ (3)$$
At first the (3) may seem ‘simple’ but of course it isn’t... an attempt will be made in next post...
Kind regards
$\chi$ $\sigma$
The non linear first order Riccati ODE...
$$ y^{\ '} = x^{2} + y^{2}\ (1)$$
... can be transformed into a linear second order ODE with the substitution...
$$y = - \frac{u^{\ '}}{u} \implies y^{\ '} = - \frac{u^{\ ''}}{u} + (\frac{u^{\ '}}{u})^{2}\ (2)$$
... so that we have to engage the ODE...
$$u^{\ ''} + x^{2}\ u =0\ (3)$$
At first the (3) may seem ‘simple’ but of course it isn’t... an attempt will be made in next post...
Kind regards
$\chi$ $\sigma$
chisigma
Well-known member
Re: Riccati's equation an Bessel functions
The solution of the ODE...
$$u^{\ ''} + x^{2}\ u = 0\ (1)$$
can be found in Polyanin A.D. & Zaitzev V.F. Handbook of Exact Solutions for Ordinary Differential Equations, 2nd edition...
$$ u(x) = \sqrt{x}\ \{c_{1}\ J_{\frac{1}{4}} (\frac{x^{2}}{2}) + c_{2}\ Y_{\frac{1}{4}} (\frac{x^{2}}{2})\ \}\ (2)$$
... where $J_{\frac{1}{4}} (*)$ and $Y_{\frac{1}{4}} (*)$ are Bessel function of the first and second type, $c_{1}$ and $c_{2}$ arbitrary constants. Now computing $y= - \frac{u^{\ '}}{u}$ leads us to the solution of the Riccati's equation...
Kind regards
$\chi$ $\sigma$
The solution of the ODE...
$$u^{\ ''} + x^{2}\ u = 0\ (1)$$
can be found in Polyanin A.D. & Zaitzev V.F. Handbook of Exact Solutions for Ordinary Differential Equations, 2nd edition...
$$ u(x) = \sqrt{x}\ \{c_{1}\ J_{\frac{1}{4}} (\frac{x^{2}}{2}) + c_{2}\ Y_{\frac{1}{4}} (\frac{x^{2}}{2})\ \}\ (2)$$
... where $J_{\frac{1}{4}} (*)$ and $Y_{\frac{1}{4}} (*)$ are Bessel function of the first and second type, $c_{1}$ and $c_{2}$ arbitrary constants. Now computing $y= - \frac{u^{\ '}}{u}$ leads us to the solution of the Riccati's equation...
Kind regards
$\chi$ $\sigma$
kjpatterson21
New member
- Jan 23, 2020
- 1
Where is the solution to -u'/u? I need to see the detail steps to arrive at the solution.