This may be very obvious but I am stuck trying to solve a boundary value problem. I am trying to solve the following differential equation : $$F^{3}F^{(5)}+F=1 \space (Eq.1)$$ I have been advised to write the parametrization : $\boxed{F=1+\phi}$ with $\phi(\infty)=0$ : I thus obtain an asymptotic behavior : $$\phi^{(5)}+\phi \approx 0$$ that have 5 exponential solutions of the form : $\boxed{\phi \approx exp(\omega_i \eta)}$ with $\omega_i^{5}=-1$.
I can write that : $$ \phi = A e^{\omega_1\eta} + B e^{\omega_2\eta}+C e^{\omega_3\eta}+D e^{\omega_4\eta}+E e^{\omega_5\eta} $$ Knowing that : $\phi(\infty)=0$ and that 2 of the 5 complex roots have positive real parts (giving growing exponentials) : the coefficients of these growing exponentials must be zero.
It is said that this remark is supposed to provide 2 boundary conditions to the $(Eq.1)$ but I don't see how it's related since these are "just" conditions on coefficients of $\phi$...
It seems very immediate but I don't see it for the moment.
Thank you in advance for your help.
With $F=1+ϕ$ the approximate equation for $ϕ\approx 0$ results from $$ ϕ^{(5)}+1+ϕ =(1+ϕ)^{-3}=1-3ϕ+6ϕ^2-10ϕ^3+...+\binom{n+2}{2}(-ϕ)^n+... $$ Thus the linear approximation is $$ ϕ^{(5)}+4ϕ=0 $$ As observed, the characteristic roots are $α<0$, $β,\barβ$ in the negative half-plane and $γ,\barγ$ in the positive half-plane, all on a regular pentagon around the origin.
This makes the equilibrium point a saddle point. Almost all solutions, and thus all numerical solutions except the constant one, which approach the saddle point will at some point turn away and exponentially diverge.
An approximation to a solution that is asymptotic to the saddle point can be obtained by using a stable solution of the linear approximation as far-field piece of a piece-wise defined function. This far-field approximation can be parametrized as $$ ϕ=c_1e^{αη}+c_2e^{βη}+c_3e^{\barβη} $$ or equivalently as a solution of the linear ODE $$ (D-α)(D-β)(D-\barβ)\phi=0. $$ In this form it is easy to obtain further linear conditions on the derivatives sequence as $$ (D-α)(D-β)(D-\barβ)\phi^{(k)}=0,~~~k=1,2,... $$ To "optimally" fit a numerical solution and a far-field approximation at some point $η=b$, one can impose these linear conditions on the derivatives $[ϕ(b),ϕ'(b),...,ϕ^{(4)}(b)]$ that make up the state space for the original equation. This results in 2 boundary conditions.
That using the differential equations as "connection condition" is easier than extending the system with the coefficients as free parameters and adding continuity conditions can also be seen in Non-unique solutions for ODE with boundary conditions at infinity