Linked Questions
10 questions linked to/from On equation $f(z+1)-f(z)=f'(z)$
9
votes
4
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2k
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How may I find all continuous and bounded functions g with the following property?
Find all continuous and bounded functions $g$
with :
$$\forall x \in \mathbb R, 4g(x)=g(x+1)+g(x-1)+g(x+\pi)+g(x-\pi).$$
I have posted this question here, but received no answer.
- 4,074
6
votes
1
answer
843
views
Solve $f(x)=\int_{x-1}^{x+1} f(t) \mathrm{d}t$
Solve $f(x)=\int_{x-1}^{x+1} f(t) \mathrm{d}t$.
When $f$ is a function, it looks like the only solution is $f(x)=0$. But what if we allow distributions, such as the Dirac delta?
- 163
1
vote
3
answers
2k
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Fourier Transform, for entire function
On THIS site, Alexandre used Fourier transform to solve the problem.
If we use Fourier transform, how to define it to ensure any entire function has a FT?
Classical FT is defined by
$$ \mathcal{F}[f]...
- 1,551
4
votes
2
answers
267
views
One question about a specific first-order differential equation
Find the function-constant pairs $\langle f(x),c\rangle$ that satisfy the differential equation below:
$$f'(x)=f(x+c),$$
where $c \in \mathbb{C}$ and $f(x) \in \mathbb{C}$.
I found two families of ...
- 696
4
votes
2
answers
352
views
Solving the functional equation $2f(x)=f(x+a_n)+f(x-a_n)$
Let $a_n$ be a sequence of strictly positive real numbers such that $\lim_{n \to \infty}a_n=0$. Find all functions $f: \mathbb{R} \to \mathbb{R}$ that admit primitives (i.e. there exists a function $F:...
- 331
1
vote
2
answers
424
views
Unusual Differential Equation for CDF
Consider the following differential equation
$$F(cx) = F(x) + x F'(x)$$
for $c>1$.
Does this differential equation belong to a some well known class?
Is there a way to find all the solutions $F(...
- 23
2
votes
1
answer
284
views
Eigenfunctions of an infinite summation operator
I would like to find ALL eigenfunctions to the operator, for $f$ a real function on R+*:
$f \rightarrow \sum_{1}^{\infty} f(nx)$
So to find $f$ such that: $\sum_{1}^{\infty} f(nx) = \lambda f(x)$
...
- 1,199
1
vote
1
answer
239
views
How to solve a linear algebraic complex equation in one function evaluated at different arguments?
Hello,
I am trying to solve an equation of the form
$C_1 f(k_1 z) + C_2 f(k_2 z) + C_3 f(k_3 z) + C_4 f(k_4 z) = C_5 z^2$
for $f(z)$. Everything is complex. The $C_i$'s and $k_i$'s depend on some ...
- 13
0
votes
1
answer
107
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Differential equation problem
I am stuck on a problem for a while. How do I solve $f'(x)=-kf(x-1)$? Unlike normal questions which has $f(x)$ on the RHS, this has $f(x-1)$ which has me stumped.
1
vote
0
answers
103
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Reference for numerical solutions for differential equations like $f'(x)=f(x+1)+f(x-1)$
One can solve a delay differential equation (like for example $f'(x)=f(x-1)$) if we have a function as a bounded condition (in my example we need to know $f$ on $[0,1)$) and then use a simple forward ...
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