I am studying the linear PDE:

$$ t^2\frac{\partial^3}{\partial t^3}\sum_{n=1}^\infty \Psi_n(t,s)=s^2\frac{\partial}{\partial s}\sum_{n=1}^\infty \Psi_n(t,s)+\sum_{n=2}^\infty b(n)\frac{\partial}{\partial s} \Psi_n(t,s)\tag{1}$$

where the $b(n)$ coefficients are $b(n)=\big\lbrace 2,12,24,40,60,84, \cdot\cdot\cdot\big\rbrace.$ Here $b(n)$ is related to the oblong numbers $a(n),$ because $b(n)=2a(n)$ for $n>2.$

Consider another linear PDE:

$$t\frac{\partial^2}{\partial t^2} \sum_{n=1}^\infty \Phi_n(x,t)=-x\frac{\partial}{\partial x}\sum_{n=1}^\infty \Phi_n(x,t)+\sum_{n=2}^{\infty}a(n)\Phi_n(x,t) \tag{2}$$

So far I have a kernel solution to $(2)$ and a kernel solution to $(1).$ The solution to $(1)$ is:

$$\Psi_n(t,s)=2 \sqrt{\frac{tn}{s}}K_1(2\sqrt{t ns})$$

where $K_1$ is a modified Bessel function.

The solution to $(2)$ is:

$$\Phi_n(x,t)=e^{\frac{nt}{\log x}}.$$

The PDE's $(1)$ and $(2)$ are related by the Mellin transform. I took the solution to $(2)$ and found it's Mellin transform and then used that to re-construct $(1)$.

Are the solutions I found unique? Do PDE's of this type have a name? (i.e. coupled with infinite natural number sequence).