To (Q1): Yes, see Differential Galois Theory. For example, the Galois group of the Airy equation over $\mathbb{C}$ is given by the Lie group $\operatorname{SL}_2(\mathbb{C})$.

To (Q2): This is posed too broad and depends on what you mean by relations, but I would say yes, see (Q4).

To (Q3): No idea. What would be a Diophantine differential set?

To (Q4): Yes, see Vinogradov's Diffieties. This kind of analogy can be taken very far, see for example the recent fascinating work of Kryczka, Sheshmani, and Yau:

Derived Moduli Spaces of Nonlinear PDEs I: Singular Propagations

Derived Moduli Spaces of Nonlinear PDEs II: Variational Tricomplex and BV Formalism

The references therein are also worth checking out.

To (Q1): Yes, see Differential Galois Theory. For example, the Galois group of the Airy equation over $\mathbb{C}$ is given by the Lie group $\operatorname{SL}_2(\mathbb{C})$.

To (Q2): Yes, see (Q4).

To (Q3): No idea. What would be a Diophantine differential set?

To (Q4): Yes, see Vinogradov's Diffieties. This kind of analogy can be taken very far, see for example the recent fascinating work of Kryczka, Sheshmani, and Yau:

Derived Moduli Spaces of Nonlinear PDEs I: Singular Propagations

Derived Moduli Spaces of Nonlinear PDEs II: Variational Tricomplex and BV Formalism

The references therein are also worth checking out.

To (Q1): yes, see Differential Galois Theory. For example, the Galois group of the Airy equation is $\operatorname{SL}_2(\mathbb{C})$.

To (Q2): this is posed too broad and depends on what you mean by relations, but I would say yes, see (Q4).

To (Q3): No idea. What would be a Diophantine differential set?

To (Q4): also yes, see Vinogradov's Diffieties.

To (Q1): Yes, see Differential Galois Theory. For example, the Galois group of the Airy equation over $\mathbb{C}$ is given by the Lie group $\operatorname{SL}_2(\mathbb{C})$.

To (Q2): This is posed too broad and depends on what you mean by relations, but I would say yes, see (Q4).

To (Q3): No idea. What would be a Diophantine differential set?

To (Q4): Yes, see Vinogradov's Diffieties. This kind of analogy can be taken very far, see for example the recent fascinating work of Kryczka, Sheshmani, and Yau:

Derived Moduli Spaces of Nonlinear PDEs I: Singular Propagations

Derived Moduli Spaces of Nonlinear PDEs II: Variational Tricomplex and BV Formalism

The references therein are also worth checking out.