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I am interested in the following type of $4$-th order linear PDE with $2$ variables (i.e., $x$ and $t$):

$$x^3 f_{xxxt}+ f =0$$

Does anyone know if this type of PDE already appeared in the literature? It looks quite simple.

P.S.
To clarify: the $f$-equation is a simplified equation based on a longer "$P$-equation" via a change of variable: $f =(x^2 t) P$ with a boundary condition $P(x,t=0)=1$. This BC rules out many solutions/ansatzes. This $4$th-order PDE can be solved, e.g., perturbatively in a small $t$ expansion, but I wonder how to approach this equation more generally, from (say) a geometric viewpoint.

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    $\begingroup$ Do you have any boundary conditions? $\endgroup$
    – user479223
    Commented Feb 27 at 3:25
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    $\begingroup$ Why the projective-geometry tag? I would have thought reference-request might be better in its place, if it's possible to drop. And why stochastic-differential-equations? $\endgroup$
    – David Roberts ♦
    Commented Feb 27 at 3:54
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    $\begingroup$ The substitution $x=e^y$ transforms this to a constant coefficient problem. This makes available the machinery of Fourier and Laplace transforms. What to do further really depends on what you want to know about the PDE. $\endgroup$
    – Michael Renardy
    Commented Feb 27 at 3:55
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    $\begingroup$ A two-parameter family of solutions is $f=Cx^a \exp [-t/(a(a-1)(a-2))]$ ... $\endgroup$
    – Michael Engelhardt
    Commented Feb 27 at 3:59
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    $\begingroup$ If $f$ is a solution then so is $-x^2f(x^{-1},-t)$. $\endgroup$
    – Bob Terrell
    Commented Mar 7 at 19:27

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