How Can We Eliminate Quantifiers in Differential Equations?

[mathmari]

Hey!

I have shown that a differential equation in the ring $\mathbb{C}[x]$ has a solution if at least one root of the charcteristic equation of the homogeneous equation is equal to $0$

We consider the language $\{+, \frac{d}{dx}, 0, 1\}$.

I want to eliminate the quantifier from the formula $\exists x \ Ly=f$. ($L$ is a differential operator.)

How could we do that?

 

[mmwave]

Hello! It's great to see that you are exploring differential equations in the ring $\mathbb{C}[x]$ and how to eliminate quantifiers in formulas involving differential operators. This is a very interesting topic in mathematics and has many applications in various fields.

To eliminate the quantifier from the formula $\exists x \ Ly=f$, we can use the method of quantifier elimination. This method allows us to rewrite the formula in a way that does not involve any existential quantifiers. In this case, we can use the fact that the characteristic equation of the homogeneous equation has at least one root equal to $0$.

First, we can rewrite the formula as $\exists x \ Ly=f$ as $\exists x \ (Ly-f=0)$. This means that there exists a value of $x$ for which the difference between the differential operator $L$ applied to $y$ and the function $f$ is equal to $0$. Now, using the characteristic equation, we can rewrite this as $\exists x \ (L-\lambda)^k y = 0$, where $\lambda$ is the root of the characteristic equation equal to $0$ and $k$ is the multiplicity of this root.

Next, using the fact that $\mathbb{C}[x]$ is a ring, we can rewrite this formula as $\exists x \ (L-\lambda)^k y - 0 = 0$. Finally, we can eliminate the quantifier by using the property that for any ring $R$, the formula $\exists x \ (a-x)^k b = 0$ is equivalent to $(a-x)^k b = 0$. Therefore, our final formula is $(L-\lambda)^k y = 0$, which does not involve any existential quantifiers.

I hope this helps you in your work and exploration of differential equations and quantifier elimination. Keep up the great work!