This is not a complete solution, but a suggestion on what you can try. First, note that the "planimeter" will "compute" for you integrals
$$
F(x)=\int ydx,
$$
where $y=f(x)$ is the given curve. This "computation" is in the form of a numerical output measured by a rotating wheel, but, using Paucellier-Lipkin inversor you can convert this to a linear motion which will lead to a geometric drawing of the graph of $F(x)$.

On the other hand, "Kempe's universality theorem" allows you to trace arbitrary planar algebraic curves using mechanical linkages. Kempe's original proof suffered from a number of deficiencies, its description could be found in e.g.

The deficiencies were fixed by Kapovich and Millson in *Universality theorems for configuration spaces of planar linkages* (published in Topology).

(Note that David Khudaverdyan has a computer-aided implementation of a curve-drawing algorithm for some planar curves.) Combination of planimeter and Kempe-like linkages, allows you to "compute integrals" of algebraic functions using a mechanical device. By playing with these constructions, maybe you can construct a device (impractical, of course), which will trace the curve you are interested in.

An interesting question coming out of this is if there is a generalization of "Kempe's universality" for *1-st order algebraic ODEs* (or even more general ODEs/PDEs), e.g.:

Suppose that $P$ is, say, a polynomial, function of several variables and you have an ODE of the form:
$$
P(u', u, t)=0,
$$

where $u=u(t)$ is an unknown function ${\mathbb R}\to {\mathbb R}^2$. Is there a "mechanical device" (where one would allow both mechanical linkages and "planimeter-like" gadgets, yet to be specified) which will trace solution curves of this ODE?

Edit: Indeed, Claude Shannon in

*Mathematical theory of the differential analyzer*, J. Math. Phys. Mass. Inst. Tech. 20 (1941), 337-354, doi:10.1002/sapm1941201337

proved that one can "draw" solutions of arbitrary algebraic differential equations using mechanical devices ("analog computers"). A gap in
Shannon's proof was noted and fixed by

- Marian Boykan Pour-el,
*Abstract computability and its relation to the general purpose analog computer (some connections between logic, differential equations and analog computers)*, Trans. Amer. Math. Soc. **199** (1974), 1-28 doi:10.1090/S0002-9947-1974-0347575-8.

On the practical side, there are real differential analyzers (like the one by V. Bush), built up to 1950s, which were used to solve algebraic differential equations.