When working with differential equations is it ever permissible to say that a function is itself the Laplace transform of some other function?
Context: I have a partial differential equation given by
$$ \frac{\partial n}{\partial t} = A \frac{\partial (\gamma^2n)}{\partial \gamma}$$
And I want to find a general solution for $n(\gamma,t)$ with the initial value of $n(\gamma, 0)$. The solution is given in the problem, and it has a form remarkably similar to a Laplace transform. To get this answer I would like to say that
$$n = \int_0^\infty e^{-\beta t}\,f(\gamma, \beta)\,d\beta$$
This converts the problem into a 1st order DE in $\gamma$ ($\beta$ is treated as a constant), which can then be solved to get the explicit dependence on $\gamma$ in $f(\gamma, \beta)$.
Am I allowed to do this?
Also, am I solving this in a completely counter intuitive manner, should I look at some other method? I can do this by taking the Laplace transform of $n$, but I do not end up with the form of the solution given.
EDIT: This actually seems to be a duplicate of this question