# Questions tagged [integral-equations]

This tag is about questions regarding the integral equations. 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.

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### Is differentiating both sides a valid way to solve an integral equation?

I'm trying to solve this integral equation for $P(k)$: $\frac{(\int_0^\infty k^{N+1}P(k)dk)^2}{\int_0^\infty k^{N-1}P(k)dk\int_0^\infty k^{N+3}P(k)dk}=1$ Not being familiar with integral equations, I'...
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### Changing a double integral into a single integral - Volterra-type integral equations

I have a question regarding a calculation that i stumbled upon when proving that a Cauchy problem can be converted in a Volterra-type integral equation. Specifically, this equality: \begin{equation*} \...
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### Solve the integral equation $\int_0^{-x}f(x')dx'= f(x) + x$. [closed]

Like it says, I'm playing around with even and odd functions and require a function such that $$\int_0^{-x}f(x')dx'= f(x) + x\,.$$ I can't think how to go about it, any help appreciated.
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### Existence of the integral equation solution

In the following equation: $$\int_{-\infty}^\infty g(x) \frac{1}{\sqrt{2\pi\sigma^2}}e^{-(x-\mu)^2/(2\sigma^2)} dx = \frac{1}{\mu^2 - 1} ,$$ The function $g(x)$ is unknown and doesn't depend on $\mu$....
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### Is it possible to find a solution to this equation, reducing it to the Fredholm equation?

To continue the questions: https://math.stackexchange.com/questions/3743374/approximation-of-the-convolution-operator?noredirect=1#comment7697800_3743374 https://math.stackexchange.com/questions/...
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### Integral equation $\frac{\phi(x)}n = \binom n {nx}\int_0^1 q^{nx}(1-q)^{n(1-x)}\phi(q)\,dq,\quad \forall x\in[0,1],$

In Sewall Wright's Evolution of Mendalian Population, the equation for the nonrecurrent mutation is $$\frac{\phi(x)}n = \binom n {nx}\int_0^1 q^{nx}(1-q)^{n(1-x)}\phi(q)\,dq,\quad \forall x\in[0,1],$$ ...
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### Integral equation problem

Consider the integral equation $\phi(x)-\frac{e}{2} \int_{-1}^{1} x e^{t} \phi(t) d t=f(x) .$ Then there exists a continuous function $f:[-1,1] \rightarrow(0, \infty)$ for which solution exists there ...
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### Solution to the following Fredholm integral equation?

I have the following integral equation $$1 = \int_{-L}^{L} K(s-t)f(t)\mathrm{d}t,$$ where we are solving for $F = \int_{-L}^{L}f(s)\mathrm{d}s$. That is, $f(t)$ is an unknown function that we do not ...