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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.

• Anupam Saxena, Kempe's Linkages and the Universality Theorem, Resonance – Journal of Science Education 16 Issue 3 (2011) pp 220-237. The deficiencies were fixed here.

(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.

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.

• Anupam Saxena, Kempe's Linkages and the Universality Theorem, Resonance – Journal of Science Education 16 Issue 3 (2011) pp 220-237.

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.

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.

• Anupam Saxena, Kempe's Linkages and the Universality Theorem, Resonance – Journal of Science Education 16 Issue 3 (2011) pp 220-237. The deficiencies were fixed here.

(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.

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.

• Anupam Saxena, Kempe's Linkages and the Universality Theorem, Resonance – Journal of Science Education 16 Issue 3 (2011) pp 220-237. The deficiencies were fixed here.

(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.

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. Original Kempe's proof suffered from a number of deficiencies (its description could be found e.g. here). The deficiencies were fixed here.  (Note that David Khudaverdyan has a computer-aided implementation of a curve-drawing algorithm for some planar curves, see here.) 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, Shannon in "Mathematical theory of the differential analyzer" J. Math. Phys. Mass. Inst. Tech. 20 (1941), 337-354, 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 M.B. Pour-el, see here.

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

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.

• Anupam Saxena, Kempe's Linkages and the Universality Theorem, Resonance – Journal of Science Education 16 Issue 3 (2011) pp 220-237. The deficiencies were fixed here.

(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.