## For any two regular languages A, B, show that {xy|x ∈ A, y ∈ B, |x| = |y|} is context-free

Basically I’m wondering if the concatenation of two equal length string is context free. I’ve seen multiple proofs of this online using PDAs but we aren’t covering them in my automata course and my professor says they aren’t needed for the proof. Any help would be greatly appreciated!

This is the full question:

For any two languages $$A, B$$ over $$\Sigma$$, define $$A \diamond B := \{xy\mid x \in A, y \in B, |x| = |y|\}$$. Show that if $$A, B$$ are regular then $$A \diamond B$$ is context-free

## Is it true that $\text{min}\{x/2, y/2\} = \frac{1}{2}\text{min}\{x, y\}$?

Is it true that $$\text{min}\{x/2, y/2\} = \frac{1}{2}\text{min}\{x, y\}$$? Or more generally, $$\text{min}\{cx, cy\} = c \cdot \text{min}\{x, y\}$$?

I think that the answer is YES. But, I got this guess by just plugging in a few numbers. Maybe I am missing some sort of clever counterexample. Can someone please help me?

## How to solve this ODE ?$(x^2) y”” + (3x^2-2x)y”’ + (3x^2-4x+2)y” +(x^2-2x+2)y’ = 0$ ( hint )

$$(x^2) y”” + (3x^2-2x)y”’ + (3x^2-4x+2)y” +(x^2-2x+2)y’ = 0$$

i tried to Guess a solution and use the fact that i can decrease the ODE to less power ( to $$y”’$$ ) by using the Wronskian .

i Guessed that $$e^{-x}$$ is a solution . is it the way we solve this kind of equations ? its homework question so i guess i can solve it.

euler ODE doesn’t work here .

## Solving for $y” – 4y’ – 5y – 2 = 0$

I am looking to solve for the above nonhomogeneous ODE. I know how to find the general solution for the reduced equation of the homogeneous form, that is, $$y” – 4y’ – 5y = 0.$$

The characteristic equation is $$r^{2} – 4r – 5 = 0$$, which gives two real and distinct roots $$r=-1,5$$.

So the complementary solution is $$y_{c}=c_{1}e^{5x} + c_{2}e^{-x}$$.

Now I am looking to guess the particular on the right-hand side but I am not sure about how to do that in order to find the general solution of the above nonhomogenous ODE.