I am trying to solve the following ordinary differential equation:
$$ \nabla^4 f(r) + \nabla^2 f(r) + kf(r) =0 \tag 1$$
where $\nabla^4$ is the radial biharmonic operator, $\nabla^2$ is the radial Laplace operator, $k$ is a constant and $r$ is the dimensionless radial coordinate. Because I was not able to find the general solution to equation $(1)$ for some time now, I decided to take a gradual approach by solving what to me seemed subproblems of my problem.
I solved the following set of subproblems (differential equations):
$$ \nabla^2 f(r) = 0 \Rightarrow f(r) = C_0 + C_1 \ln(r) \tag 2$$ $$ \nabla^4 f(r) = 0 \Rightarrow f(r) = C_0 + C_1 \ln(r) + C_2r^2+C_4 r^2 \ln(r)\tag 3$$ $$ \nabla^2 f(r) + kf(r)= 0 \Rightarrow f(r) = C_0J_0(\sqrt{k} \cdot r) + C_1 Y_0(\sqrt{k} \cdot r) \tag 4$$
$$ \nabla^4 f(r) + \nabla^2 f(r) = 0 \Rightarrow f(r) = C_0 + C_1 \ln(r) + C_2J_0(r) + C_3 Y_0(r) \tag 5$$
However, I still do not know how to find the general solution to equation $(1)$, or even to the equation:
$$ \nabla^4 f(r) + kf(r) = 0 \tag 6$$
which I also consider a subproblem.
Currently, I am considering whether this approach can solve my problem. Consider the following equation:
$$ \nabla^4 f(r) + \nabla^2 f(r) = g(r) \tag 7$$
The general solution to the above equation would be:
$$ f(r) = C_0(r) + C_1(r) \ln(r) + C_2(r) J_0(r) + C_3(r) Y_0(r) \tag 8$$
My plan is to try to use the variation of constants method to solve equation $(7)$, with the constraint $g(x) = -kf(x)$. This way, I would be actually solving equation $(1)$. My question is, is this a valid way to solve equation $(1)$? If not, is there some other approach I can take?