Tagged Questions

3
votes
2answers
58 views

If $f$ is even and $y'=f(y)$ then $y$ is odd

Let $f\in C^1(\mathbb{R}, \mathbb{R})$ be an even function. Consider the maximal solution $y\colon\left]\alpha ,\beta\right[\to \mathbb{R}$ of the IVP $$y'=f(y),\ y(0)=0$$ Prove that $y$ ...
1
vote
1answer
41 views

Second order nonlinear delay differential equation

I have to solve the following delay differential equation: $$\ddot{x}(t)+A\sin(\omega x(t-\tau))=0$$ Can someone give me a hint on how to solve this equation? Thanks
0
votes
1answer
37 views

Finding the Extremals of a Functional J.

The functional $J$ is defined on smooth functions $y \colon [a,b] \to \mathbb{R}$ satisfying $y(a) = u$, $y(b) = v$ and is given by $$J[y]=\int_a^b \sqrt{y} \sqrt{1+(y')^2}\, dx.$$ I have found ...
11
votes
1answer
143 views

How find this function $f(x)$

Let $f:\mathbb{R}\to\mathbb{R}$ be continuous such that $\dfrac{f(x)}{x}=f'\left(\dfrac{x}{2}\right)$. Find $f(x)$. (2):if $f(x)$ on $x\in[a,b] $ be continuous,find all $f(x)$? I think this is an ...
13
votes
3answers
263 views

Solving Differential Functional Equation $f(2x)=2f(x)f'(x)$

Find all functions satisfying $f(2x)=2f'(x)f(x)$ Under given condition, can't we find explicit solutions?
1
vote
1answer
29 views

If $h(x,c)$ is solution to equation $f'(x)=g(f(x))$, what is the solution for $-f'(x)=g(f(x))$?

Assume that we have differential equation $$f'(x)=g(f(x))$$ and we know that function $h(x,c)$ is the solution (with parameter) to it, then what can be said of the solution for the equation ...
1
vote
3answers
93 views

Who can solve this ordinary differential equation?

$$f'=f(1-x)$$ This equation appears when I try to solve the eigenvalue problem of an integral equation.
3
votes
2answers
65 views

Solution to functional equation

I have the following functional equation: $$f(a+b)=f^{n-1}(a)f(b)+f^{n-2}(a)f^{1}(b)+...+f(a)f^{n-1}(b)=\sum_{k=0}^{n-1}f^{n-1-k}(a)f^{k}(b)$$ where $a,b$ are complex and the function $f$ is an ...
4
votes
2answers
69 views

The number of non-trivial polynomial solutions of the differential equation $x^3y'(x)=y(x^2)$

I came across the following problem that says: The number of non-trivial polynomial solutions of the differential equation $x^3y'(x)=y(x^2)$ is which of the following? $(1)0\space (2)1 ...
1
vote
1answer
42 views

If $f(cx, cy) = f(x, y)$ can we always find $g$ such that $g(\frac{x}{y}) = f(x, y)$?

Motivation: A differential equation $\frac{dy}{dx} = f(x, y)$ is said to be homogeneous if we can find $g$ such that $g(\frac{x}{y}) = f(x, y)$, which allows us to solve the equation using the ...
0
votes
0answers
36 views

differential inequality : $1+f(x) > Cxf'(x)$, and $f(x)= f(1/x)$

I am working on a differential equation which has the following conditions: $f$ is defined from $(0,\infty)\to \mathbb{R}^+$. $f$ is differentiable. $f(x) = f(1/x)$ for all $x$. $f$ is increasing ...
0
votes
0answers
61 views

What kind of (differential) equation is this?

This may be a silly question, but I am confused with the following. To my knowledge, in general any initial value problem we have a differential equation of the form $\dot{y}(x)=f(x,y(x))$ plus an ...
1
vote
0answers
55 views

Functional equation for the given function

For instance, there is functional equation for Lambert W function $z=W(z) e^{W(z)}$ And moreover, there is differential one: $z(1+W)\frac{dW}{dz}=W$. At the same time, there is no known functional ...
0
votes
0answers
50 views

properties about number of groups of linearly independent solutions in the general solutions of linear functional-differential equations

Are there any effective methods to determinate the number of groups of linearly independent solutions in the general solutions of linear functional-differential equations of the form ...
0
votes
1answer
206 views

functional equation uniqueness $f(x^{2})= f(x)^{2} $ in $\mathbb{C}$ and $f(x)+f''(x) = 0 $ in $\mathbb{C}$

This is an exercise from a book I tried: One would like to find all holomorphic equations that satisfy:$$i) \ f(z)+f''(z) = 0 \text{ in } \mathbb{C} $$$$ii)\ f(z^{2})=f(z)^{2} \text{ in } ...
6
votes
3answers
265 views

Solving $\forall x \in \mathbb{R}_+^*, f'(x) = f\left(\frac1{x}\right)$

I recently came across this equation : $$\forall x \in \mathbb{R}_+^*, f'(x) = f\left(\frac1{x}\right)$$where $f \in \mathcal{C}^1(\mathbb{R}, \mathbb{R})$. I've done the following, but I'm stuck at ...
3
votes
1answer
154 views

A differential-functional equation: $f'(f^{-1}(x)) = 1/g(x)$

Problem: Given $g(x)$, solve the equation $f'(f^{-1}(x)) = \frac{1}{g(x)}$ for an invertible and differentiable function $f(x)$. So far I have tried setting $y = f^{-1}(x) \Leftrightarrow x ...
5
votes
1answer
194 views

How to solve DE that relate values of derivatives at different points?

I try to solve for the specific function $f(x) = \frac{2-2a}{x-1} \int_0^{x-1} f(y) dy + af(x-1)$ It looks similar to the function used to find the Renyi's parking constant because it came out from a ...