# Tagged Questions

An integral equation is an equation in which the unknown function appears under the integral sign. There is no universal method for solving integral equations. Solution methods and even the existence of a solution depend on the particular form of the integral equation. (Handbook of Mathematics - ...

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### Why are mathematician so interested to find theory for solving partial differential equations but not for integral equation?

Why are mathematician so interested to find theory for solving partial differential equations (for example Navier-Stokes equation) but not for integral equations?
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### if $f(x)-a\int_x^{x+1}f(t)~dt$ is constant, then $f(x)$ is constant or $f(x)=Ae^{bx}+B$

Question: let $a\in(0,1)$, and such $f(x)\geq0$, $x\in R$ is continuous on $R$, if $$f(x)-a\int_x^{x+1}f(t)dt,\forall x\in R$$ is constant, show that $f(x)$ is constant; or ...
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### Solve these functional equations: $\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$

Let $f : [0,1] \to \mathbb{R}$ be a continuous function such that $$\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$$ Determine all such ...
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### 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 - ...
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### Eigenvalues and eigenfunctions for the Fredholm integral operator $K(g) = \int_0^1 e^{x t} g(t) dt$.

I would like to compute the eigenvalues and eigenfunctions for the Fredholm integral operator $$K(g) = \int_0^1 e^{xt} g(t) dt.$$ The sources I've checked* seem to say that the process is fairly ...
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### How to solve Fredholm integral equation of the second kind? ($f(x) - \lambda\int\limits_0^1 9xtf(t) dt = ax^2 - 4x^2$)

I have an equation : $\ f(x) - \lambda\int\limits_0^1 9xtf(t) dt = ax^2 - 4x^2\$ in $\ L_2[0,1]\$ space. And I want to understand how to solve it, not just obtain an answer.
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### Volterra integral equation of second type solve using resolvent kernel

Solve the integral equation $$y(t)= f(t) + \lambda \int_{0}^{t} (t-s) y(s) ds$$ where $f$ is continuous using the method of finding the resolvent kernel and Newmann series. Here it is what I did:...
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### Integral equation solution (Fredholm, second type)

There is an equation $$w(x) = g(x)+\int\limits_0^M w(y)f(x-y)\,dy$$ where $f\geq 0$, $f\in C^\infty(\mathbb R\setminus\{c\})$ for some point $c$ and $\int\limits_{-\infty}^\infty f(t)\,dt\leq 1$. ...
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### How to solve following differential equation?

$$\int \limits_{0}^{\infty}\sqrt{1 + y'^{2}(x)}dx = 2 \sqrt{x} + y \qquad (.1)$$ The solution is $$3y = x\sqrt{x} - 3\sqrt{x} .$$ I don't know how to solve this type of equations. Also I don't ...
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### 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 $\dfrac{d}{dx}f(x)^2=2f(x)f'(x).$ ...
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### Anomalous integral equation

I'm trying to solve the following equation: $$\int_0^{f(x)}f(t)dt=g(x)$$ Differentiating under integral I obtain: $$f[f(x)]\frac{d}{dx}f(x)=\frac{d}{dx}g(x)$$ I know the function $g(x)$. Is there a ...
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### Differential equation with shifited term

I have a differential equation (Or integral equation) of the form: $$f(x) = a e^{-x} + b \int_0^x f(cz+dx) e^{-z} dz$$ $a,b,c,d$ are constants. I am considering whether the above equation has a ...
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