I am trying to solve this non linear differential equation with 3 boundary conditions :
$$ R\dfrac{d^3 h}{d R^3}+2\dfrac{d^2 h}{d R^2} + k\dfrac{d h}{d R} = 0$$
With : $\frac{d h}{d R}(R=0)=0$, $h(R=L) = \frac{d h}{d R}(R=L)=0$
Unfortunately, I am stuck because I haven't been able to find a reliable way to solve it correctly.
How do you think it would be possible to find an analytical solution to this equation ?
Thank you,
Cheers.
Since you have derivatives of $h$ of orders $1$, $2$, and $3$ (but not $0$, namely $h$ itself), we can make the substitution $y = \frac{dh}{dR}$, resulting in the second-order linear equation in $y$: $$ R\, \frac{d^2 y}{dR^2} + 2\, \frac{dy}{dR} + k\, y = 0. $$ We can multiply through by $R$, yielding: $$ R^2\, \frac{d^2 y}{dR^2} + 2R\, \frac{dy}{dR} + kR\, y = 0. $$ Now the first two terms are a derivative of a product: $$ \frac{d}{dR} \biggl( R^2\, \frac{d y}{dR} \biggr) + kR\, y = 0. $$ This is the standard form for a (homogeneous) Sturm–Liouville equation. There is a robust theory for studying solutions to equations of this form, however you're probably not going to see an analytic solution in a closed form unless you're willing to use Bessel functions and other special functions.
You can, of course, find a series solution, starting with the assumption that $$ y = \sum_{n=0}^\infty a_n R^n, $$ and using the differential equation in its original form to determine recurrence relations among the coefficients (and use the boundary values to get initial values).