How to numerically solve this nonlinear ODE?u″+2ru′−K(1+r2)2u=u5with initial conditions u(0)=d&gt;0 and u′(0)=0.

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How to numerically solve this nonlinear ODE?u″+2ru′−K(1+r2)2u=u5with initial conditions u(0)=d&amp;gt;0 and u′(0)=0.

annieljcddj0 · Accepted Answer

Step 1In the limit&amp;nbsp;r→0&amp;nbsp;you get by mean value or&amp;nbsp;L&#039;Hopital3u&amp;nbsp;″(0)−Ku(0)=u(0)5.Thus for some small interval around zero your solution reads asu(r)=d+Kd+d56r2.So start the interval at some positive smallish h with initial conditionsu(h)=d+Kd+d56h2,u′(h)=Kd+d53hIf necessary prepend the&amp;nbsp;r=0&amp;nbsp;point to the solution arrays.

Videoad3u · Accepted Answer

Step 1If you&#039;re looking to investigate it numerically, perhaps try making the substitutionu1=u,u2=dudr,from which you get the systemdu1dr=u2du2dr=u15+dK(1+r2)2u1−d2ru2and then numerically solving using a Runge-Kutte method or similar approach. As mentioned by Lutz Lehmann you will have to consider the initial time step analytically via some other means due to the 1r term.

How to numerically solve this nonlinear ODE? \(\displaystyle{u}{''}+{\frac{{{2}}}{{{r}}}}{u}'-{\frac{{{K}}}{{{\left({1}+{r}^{{2}}\right)}^{{2}}}}}{u}={u}^{{5}}\) with

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