How to solve this <PERSON> equation?
I am taking an intro level quantum mechanics class. Our textbook gives a problem like this: " The deuteron is a nucleus of "heavy hydrogen" consisting of one proton and on neutron. As a simple model for this nucleus, consider a single particle of mass $m$ moving in a fixed spherically -symmetric potential $V(r)$, defined by $V(r)=-V_{0}$ for $rr_{0}$. This is called a spherical square-well potential. Assume that the particle is in a bound state with $l=0$.
(a) Find the general solution $R(r)$ of the <PERSON> equation for $rr_{0}$. Use the fact that the wave equation must be finite at 0 and $\infty$ to simplify the solution as much as possible. (You do not need to normalize the solutions).
(b) The deuteron is only just bound: ie., $E$ is nearly equal to $0$. Take $m$ to be the proton mass, $m=1.6710^{-27}kg$. and take $r_{0}$ to be a typical nuclear radius, $r_{0}=110^{-15}m$. Find the value of $V_{0}$ (the depth of the potential well) in MeV (1 MeV=$1.6*10^{-13}$J (Hint: The continuity conditions at $r_{0}$ must be used. The radial wave function $R(r)$ and its dervivative $R'(r)$ must both be continuous at $r_{0}$; this is equivalent to requiring $u(r)$ and $u'(r)$ must both be continuous at $r_{0}$, where $u(r)=rR(r)$. 474
The resulting equations cannot be solved explicitly but can be used to derive the value of $V_{0}$.
I found I need to solve an equation of this form when $r<r_{0}$:$$-\frac{\hbar^{2}}{2mr}\frac{\partial}{\partial_{r}^{2}}(rR(r))-V_{0}R(r)=ER(r)$$
After expansion this becomes a second order ODE I am not able to solve. But I am not able to use the condition $r=0$ to obtain any good results. At $r=0$ we should have $-V_{0}R(r)=ER(r)$, which implies $E=-V_{0}$. (Is this allowed?). Then I can get $$-\frac{\hbar^{2}}{2mr}\frac{\partial}{\partial_{r}^{2}}(rR(r))=0$$ yet solving this implies $rR(r)=c_{0}r+c_{1}$, which I "feel" impossible since it would imply $R(r)=c_{0}+\frac{c_{1}}{r}$, which has a singularity at $r=0$ unless $c_{1}=0$.
On the other hand at $r=\infty$ we shoul have the equation become: $$-\frac{\hbar^{2}}{2mr}\frac{\partial}{\partial_{r}^{2}}(rR(r))=ER(r)$$ Hence naming $rR(r)=S(r)$, we should have $$\frac{\partial}{\partial_{r}^{2}}S(r)=kS(r), k=-\frac{2mE}{{\hbar}^{2}}$$
and solve it we have $S(r)=e^{\sqrt{k}r}\Rightarrow R(r)=\frac{e^{\sqrt{k}r}}{r}$. This function is obviously non-constant at the origin and since $k$ is negative, it actually "blow up". I do not know how to reconcile this desperate situation. So I need some help. Hence I cannot continue to part $(b)$ as well. I feel there is something very wrong in my approach. I also want to ask what does it mean "the deuteron is only just bound", why this implies $E$ is nearly equal to $0$? Also, if I cannot solve the equation explicitly, how can I get $V_{0}$'s value? I am very confused. 955