0
votes
2answers
85 views

A Matrix Integral Equation

We have an integral equation on matrix. ${\Im(t)}=\Im(0)+\int_{0}^{t} \Im(s)[K(s)]_{ \times }ds \tag 1$ $[\hspace{.2cm} ]_{\times}$ is skew symmetric matrix with diagonals zero and is non ...
1
vote
0answers
37 views

existence and uniqueness of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
0
votes
1answer
41 views

Show this integral operator is compact for various values of $\alpha$

I am having some problems evaluating a multivariable integral. This question is features in Stakgold's book Green's functions and boundary value problems. page 359. Consider the kernel for $a\leq ...
1
vote
0answers
32 views

numerical solution of integral equation

Consider the basic type of integral equation. In particular, a volterra integral equation of the first kind. That is, we have the following integral equation $$\int_a^xf(s)g(s,x)~ds=h(x)$$ where $h$ ...
1
vote
4answers
54 views

Solve $y = 2 + \int^x_2 [t - ty(t) \,\, dt]$

While working on some differential equation problems, I got one of the following problems: $$y = 2 + \int^x_2 [t - ty(t) \,\, dt]$$ I have no idea what an integral equation is however, the hint ...
2
votes
2answers
61 views

Solving an integral using Laplace transform and inverse Laplace transform

I want to solve this integral equation using Laplace: $$ Y(t) + 3{\int\limits_0^t Y(t)}\operatorname d\!t = 2cos(2t)$$ if $$ \mathcal{L}\{Y(t)\} = f(s)$$ then, $$ f(s) + 3 \frac{f(s)}{s} = ...
0
votes
0answers
40 views

An integral equation of the first kind with variable limits

How should one solve the following integral equation to find $F(\tau)$: $$ \int_{\tau -T}^{\tau}F(t)X(\tau -t)dt=e^{-(\frac{\tau}{T})^2} $$
1
vote
2answers
33 views

Reading Speed for Constant Time to Finish

You open a very long new book on your e-reader and read a few pages. It helpfully informs you that based on your reading speed you have 16 hours of reading left until you are done. You read the rest ...
1
vote
1answer
56 views

Integral equation question

If f(x) and f(t) both have the same domain and range, is there a general way to find $\int_{0}^{x^2} f(t) dt = f(x)$ given t? The actual problem tells that t = 9 and f(x) = $5 e \exp{x cos ...
5
votes
1answer
188 views

Find all continuous functions such that $f(x)^2 = \int_0^x f(\xi)\frac{\xi}{1+\xi^2}d\xi.$

From Spivak Find all continuous functions such that $$f(x)^2 = \int_0^x f(\xi)\frac{\xi}{1+\xi^2}d\xi.$$ $f(x)^2$ is clearly differentiable. I'd like to write $\dfrac{d}{dx}f(x)^2=2f(x)f'(x).$ ...
3
votes
2answers
66 views

Find all functions such that $f'(t) = f(t) + \int_0^1 f(\tau)\,d\tau.$

From Spivak Find all functions such that $f'(t) = f(t) + \int_0^1 f(\tau)\,d\tau.$ My approach: differentiate both sides to get $f''(t) = f'(t)$, giving $f'(t) = Ce^t$, implying $f(t) = Ce^t + ...
0
votes
2answers
73 views

Question about integral is equal to zero

Suppose we have the following equation $$ \int_0^\infty {f(x,r)g(x)dx} = 0 \quad {\rm for \, all}\, r\in \mathbb{R} $$ where the function $g(x)$ does not depend on $r$, while $f(x,r)$ is function of ...
2
votes
1answer
173 views

Differentiating a convolution integral

I'm trying to turn the integro-differential equation $\phi'(t) + \phi(t) = \int_0^t \sin{(t - \xi)} \, \phi(\xi) \, \mathrm{d} {\xi}$ into the differential equation $\phi'''(t) + \phi''(t) + ...
2
votes
1answer
142 views

How to differentiate this equation involving an integral expression?

I want to differentiate the Volterra integral equation $\phi(t) + \int_0^t (t - \xi) \, \phi(\xi) \, \mathrm{d}{\xi} = \sin{2t}$. Am I right in thinking that the integral can just be removed like so? ...
0
votes
0answers
48 views

System of integral equations for a unimodal symmetric probability distribution

Let $f(x)$ be a symmetric unimodal probability distribution on $\mathbb R$, with mean $\mu=0$. By unimodal, I mean that $f(x)$ is strictly increasing for $x<\mu$ and strictly decreasing for ...
7
votes
0answers
116 views

How to solve this integral equation?

Solve this integral equation: $$ {{\rm e}^{{\rm i}k\,\sqrt{\vphantom{\Large A}\,r^{2} + z^{2}\,}\,} \over \sqrt{\vphantom{\large A}r^{2} + z^{2}\,}} = \int_0^{\infty}{\rm K}_{0}\left(\lambda r\right) ...
0
votes
0answers
47 views

Integral Functional Equation

If $f(x) = 1 + \int_{x/c}^1 f(t) dt$, find $f(x)$ in terms of $c$. This problem was inspired by trying to find the expected value of the longest increasing subsequence $a_1, a_2, a_3, \cdots, a_n$ ...
13
votes
4answers
363 views

Find a continuous function $f$ that satisfies…

Find a continuous function $f$ that satisfies $$ f(x) = 1 + \frac{1}{x}\int_1^x f(t) \ dt $$ Note: I tried differentiating with respect to $x$ to get an ODE but you get one that contains integrals - ...
-2
votes
2answers
122 views

Integral equation that's cant solve… Need a hand [closed]

Help me solve this integral equation, I'm having some troubles... I need to use the Fredholm method for second kind integral equations. $$\phi(x)= \sin(x)+ \lambda \int_{0}^{\pi}\cos(2x+y)\phi ...
-4
votes
2answers
297 views

Integral eigenvectors and eigenvalues

I need to find the eigenvalues e eigenvectors of this integral. a) $$\int_{0}^{2\pi}(\cos^2(x+y)+1/2)\phi (y)dy$$ b)- Solved thanks $$\int_{0}^{1}(x^2y^2-2/45)\phi (y)dy$$
1
vote
2answers
463 views

Find the eigenvalues and eigenvectors of an integral operator

I need to find the eigenvalues e eigenvectors of this integral. $$\int_{0}^{1} K(x,y)\phi (y)dy,$$ where $K(x,y)=x(1-y),\; 0 \le x\le y \le 1$ and $K(x,y)=y(1-x),$ $0\le y\le x \le 1$ I ...
4
votes
2answers
84 views

Integration solving problem

A integration is given $$x-x_0 = \pm \int_{0}^{\phi(x)}\frac{d\Phi}{\sqrt\frac{\lambda}{2}(\Phi^2-\frac{m^2}{\lambda})} \tag{1}$$ The author said that, equation (2) can be written from equation (1) by ...
1
vote
1answer
152 views

Solve the integral equation

$$y(x) = 2 + \int_8^x (t-ty(t))dt$$ I am having a very hard time doing this problem. (i) Solve the separable differential equation $$y'(x) = x − xy(x)$$ to get $$y(x) = 1 + c \cdot e^{−x^2/2}$$ (ii) ...
6
votes
2answers
163 views

If $f(x)+f'(x)-\frac{1}{x+1}\int_{0}^{x}f(t)dt=0$ and $f(0)=0$, then what is $f'(x)$?

$f\in C^{1}[0,\infty)$, $f(0)=0$ and $$ f(x)+f'(x)-\frac{1}{x+1}\int_{0}^{x}f(t)dt=0 $$ then $f'(x)=$ ? I'v tried in the following ways. First, let $F(x)=\int_{0}^{x}f(t)dt$, then we are left to ...
0
votes
3answers
120 views

Find a $f(x) \not=0$ satisfies $\int_1^{\infty}(1-\frac{1}{x})f(x)dx=0$

can we find a function $f(x)\not=0$,such that $$\int_1^{\infty}\left(1-\frac{1}{x}\right)f(x)dx=0$$ who can give an instance ? thanks
2
votes
1answer
283 views

Difficult integral equation with function

Suppose $f(x)\in C^1([0,1])$ and $f(0)=0$. Let $$\phi(x)= \begin{cases} \int_0^x\frac{f(t)}{\sqrt{x-t}}dt &\quad\text{if}\quad x\in(0,1]\\ 0&\quad\text{if}\quad x=0 \end{cases} $$ (a) Prove ...
9
votes
2answers
351 views

Solving for unknown functions

I am not a mathematician, so excuse if my question is silly or badly stated. I have the following problem. I have 2 conditions on two unknown continuously differentiable functions: ...
2
votes
2answers
209 views

Evaluate the definite integral: $y(x) = \int_{0}^{\pi} \sin(x+y(x)) dx$

We were recently asked to evaluate this - $y(x) = \int_{0}^{\pi} \sin(x+y(x)) dx$ I think we can start by breaking up the integral as $y(x) = \int_{0}^{\pi} \sin(x)\cos(y(x)) dx + \int_{0}^{\pi} ...
1
vote
4answers
313 views

Finding all functions $f$ satisfying $f'(t)=f(t)+\int_a^bf(t)dt$

I am trying to find all functions f satisfying $f'(t)=f(t)+\int_a^bf(t)dt$. This is a problem from Spivak's Calculus and it is the chapter about Logarithms and Exponential functions. I gave up ...
3
votes
1answer
105 views

Find $f(x)$ such that $f(x) = 1 + \frac1{x} \int_1^x f(t) \mathrm{d}t$. What am I doing wrong?

I need to find a continuous function defined for real and positive $x$ such that $f(x) = 1 + \frac1{x} \int_1^x f(t)\ \mathrm{d}t$. What I did is the following: $$\begin{align*}f(x) &= 1 + ...
6
votes
1answer
159 views

Existence and Uniqueness of a solution

I'm stuck with this problem so hopefully somebody will help me :) Here you are: Let $K\in C([0,2])$ be positive, decreasing and such that $K(0)=1$. Prove that for every $h\in C([0,1])$ there exists a ...
3
votes
1answer
110 views

Finding $f(x)$ from an equation consisting of integrals

I worked out a solution for a problem I am studying for an entrance exam, but I am not sure if it is correct. I would appreciate if some knowledgeable person could help me out. Given $$ \int_0^x ...