I need a function that abides by this definition: $F(x+1)-F(x)=x$

Question

I am trying to find a solution to the differential equation: $$\frac{\mathrm{d}y}{\mathrm{d}x}=\lfloor{x}\rfloor.$$ And one solution is $$F(x+1)-F(x)=x.$$ I do not know any strategies to determine the definition of the function $F(x)$. I have seen that some Riemann functions have some functional definitions like this, and I was wondering if anyone knows a function that has the definition: $F(x+1)-F(x)=x$.

Answer 1

Your function is $$ f(x) = (1/2)x(x-1)$$

Notice that $$ f(x+1)-f(x) = (1/2) x (x+1) - (1/2) x(x-1) =x$$

The way that I came up with this form was to differentiate the equation twice and notice that $$ f''(x+1)=f''(x).$$

Thus $$f''(x)=C$$ which implies that $ f(x)$ is a polynomial.

Find the coefficients from the given equation and you have the solution.

Answer 2

Can't you just solve it separately on each interval?

As I understand it, on each interval $[a,a+1)$ the DE read $$\frac{dy}{dx}=a$$ which has solution $y = ax, x \in [a,a+1).$ And you can add a constant to get a new solution of course.

So now just patch them together to get something continuous:

• If $x \in [0,1)$ then $y = 0$.
• If $x \in [1,2)$ then $y = x-1$.
• If $x \in [2,3)$ then $y = 2x-4$
• etc.