# 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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### How to solve this special case of Fredholm Integral Equation of the First Kind

General form of 'Fredholm Integral Equation of the First Kind' $f(x) = \int_a^b{K(x,t)\phi(t)} dt$ Where $\phi(t)$ is the unkown My special case is $1 = \int_a^b{k(t)\phi(t)} dt$ A trivial ...
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### How can I solve this integral equation by converting it to a differential equation

Let we have the following integral equation :$$y(x)=e^{-x}cos(x)-\int_{0}^{x}e^{-x+t}cos(x)y(t)dt$$ How can I solve this integral equation by converting it to a differential equation
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### Solve integral equation by converting to differential equation

One has the following integral equation: $$y(x)=\frac{1}{1+x^2}+\int_{0}^{x}\sin(x-t)y(t)dt$$ How can I solve this integral equation by converting it to a differential equation?
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### prove $\int_0^1 (1-x^p)^\frac{1}{q} \,dx = \int_0^1 (1-x^q)^\frac{1}{p} \,dx$

prove that for every $p,q \gt 0$ $$\int_0^1 (1-x^p)^\frac{1}{q} \,dx = \int_0^1 (1-x^q)^\frac{1}{p} \,dx$$ I tried to start from one side and change variables to get something similiar to the ...
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### A Gronwall type inequality for Fredholm operators

I am interested in finding out the result of Gronwall type inequality for Fredholm operators. What will be the form? How one can show such inequality for fredholm operators.
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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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### Integral Equation Unknown Limits

What is the name of an equation, where the unknown is one of the limits of integration? Is there a theory that studies such equations, standard methods of solution? The simplest example is the ...
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### Is there a solution to this integral equation?

Consider the following equation $$H(y) = \int_{0}^{\infty} G\left(\frac{y-\phi_2(v)}{\phi_1(v)} \right) \exp(-v) ~\mathrm{d}v$$ where $\phi_i$ are well-behaved differentiable functions on the ...
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