Deriving the IDE we get

$$
-n b\psi(b) + c_0 \psi''(b) + (b^2+k)\psi'(b) + 2b\psi(b) = 0
$$

or

$$
c_0 \psi''(b) + (b^2+k)\psi'(b)+b(2-n)\psi(b) = 0
$$
 Now if $n < 2,\ \ c_0 > 0,\ \ k > 0$ it looks as an stable ODE.