I have a heterogeneous modified Bessel equation: $$z'' +\frac{1}{r}z'-az=B(r),$$ and I know its extremum conditions: $z'(r=0)=0$ and $z''(r=b)=0,$ and boundary conditions:$\lim_{r \to +\infty} z(r)=0$.
Can I apply this condition to my ODE without solving it? If yes how do I apply boundary conditions?
If I am interested in the values of a function at its extremum. can I put the conditions of this extremum in the general equation and then solve it thereby find the values of the function.
For example: I'm only interested in the value of the function at r = 0, and accordingly only the condition that the first derivation is zero. Can I make: $z'' -az=B(r)$? Solve it with a method variation of parameters, and apply on this my boundary condition? Probably boundary conditions I can not apply, because they only try at z(r=0), and I have at z(r->inf).
Another questions, if it's possible to apply both conditions, and I get two equations with two unknown parameters.
for r=0: $$z'' -az=B(r)$$
for r=b: $$\frac{1}{r}z'-az=B(r),$$
solve both equation separatly, and compare both solution.
Generally you cannot enforce $n+1$ conditions on the solution of an ODE with order $n$. Enforcing just $z'(0)=0$ and $z''(b) = B(b)-\frac{1}{b}z'(b)-az(b)=0$ makes the problem effectively a boundary value problem, which has a unique solution defined $r\in[0,b]$ more likely than not, however there is likely no guarantee of any behavior outside of that interval. Some rough asymptotics seem to indicate that as $r\to\infty$, $z\to 0$ if $B\to0$ and $a>0$ so maybe you can get lucky in these regimes.