# 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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### Solve $\int_0^T f(t) dt =1$ for T.

I have to solve this equation for a physics problem and I don't know where to start: $$\int_0^T f(t) dt =1 \quad\text{and}\quad f(T)=C$$ Where $T>0$, $C>0$ and $f(t)>0$ we can suppose that ...
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### Help me identify these sorts of equations

$$\int^x_0f(t)\,dt = xe^{2x}+\int^x_0e^{-t}f(t)\,dt$$ Assume $f$ is continous, solve for $f$. NB! I'm in my first calculus course so nothing too advanced please. While searching for a name for ...
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### Integral equation $f(x) - \int_0^x f(t)dt = 0$

I'd like to know the solution (if solvable) to the following integral equation $$f(x) - \int_0^x f(t)dt = 0$$ Also I'd like to know what is the required mathematical background to be able to find a ...
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### Existence and uniqueness of Volterra integral equations of the first kind with vanished kernel [duplicate]

$$\int_0^t k(s,t)f(s)ds=g(t)$$ To prove the existence and uniqueness of solutions of Volterra integral equation(VIE) of the first kind, we usually differentiate it and convert to the VIE of the ...
2answers
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### Finding function $f(x)$

How do we find the function(s) $f(x)$ given that $$f(x)=\int_{0}^{x} te^tf(x-t) \ \mathrm{d}t$$ My Try : I first used the property $\int_{0}^{a}g(x) \ \mathrm{d}x=\int_{0}^{a}g(a-x) \ \mathrm{d}x$ ...
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### Volterra equation for a Bessel type IVP that appears in inverse scattering

I have the following i.v.p. (Colton Kress-Inverse acoustic and electromagnetic scattering theory, Springer) $$y''(r)+(k^2n(r)-\frac{l(l+1)}{r^2})y(r)=0$$ $$y(0)=0, y'(0)=1$$ using the Liouville ...
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### Integral equation involving magnitude/modulus squared

I wish to solve the following integral equation that has popped up in my studies of focused light. If you notice, it looks almost like a homogenous Fredholm integral equation of the first kind. ...
1answer
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### Define all functions using the main statement

Define all functions that are continious and fullfill the equation $$f(x) = -1 + \int_0^{x^2} \frac{(f(\sqrt{t})^2 \sin t}{\cos^2t} dt$$ I'm completely lost on this one. I think that you should ...
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### Power series solution to integral equation

Hi guys i'm reading a paper in which the authors have two coupled integral equation for the function $f(x)$ and $g(x)$, in order to solve this problem they employ a power series expansion of these ...
1answer
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### Power series function expansion as solution for integral equation

I'm facing an integral equation whose unknown is a function $f(x)$: The equation is of the kind: $$K = \int_{-l}^{l} G(x,s)f(s)ds$$ So it's a Fredholm integral equation that is rewritten in this ...
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### 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 $h$ ...
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### Prove that some function is the solution of some equation

Show that $$x(t)=\sum_{n=0}^{\infty}\frac{(-1)^n(t/2)^{2n}}{(n!)^2}$$ is the solution of $$x*x=\int_{0}^t x(u)x(t-u)du=\sin t$$ My approach: I suppose that I have to use the Laplace transform. I ...
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### What functions satisfy $\int_{-\infty}^{\infty} f(x,y)dx=\int_{-\infty}^{\infty} g(x,y)dx$?

What functions satisfy $\int_{-\infty}^{\infty} f(x,y)dx=\int_{-\infty}^{\infty} g(x,y)dx$ for all $y\in \mathbb{R}$? Under what conditions would this imply that $f(x,y)=g(x,y)$?
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### Integral equation/ODE

I have to find all the functions $f(x)$ such that $$f(x)=xe^{(1-x^{2})/2}-xe^{-x^{2}/2}\int_{1}^{x}t^{-2}e^{t^{2}/2}dt$$ which satisfies $$f(x)=1-x\int_{1}^{x}f(t)dt$$ I tried to equal both, but when ...
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### Fredholm Integral in Bayesian Appliation

Let $X = x_1, x_2, \ldots, x_n$ be a sequence of Bernoulli random variables with $k$ successes. Suppose that, given $X$, the posterior predictive probability of $x_{n+1} = x$ is known to be $g(x)$ ...
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### 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 ...
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### Solving integral equation using Laplace transform

$x(t)+\int_0^t(t-\tau)x(\tau)=t^2$ Is $x(\tau)$ the equivalent of $d\tau$? How do I solve this particular equation?
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### Finding the integral of $\cos\theta \cdot dt$ in terms of the integral of $\sin\theta \cdot dt$

I have an integral as follows: $$\int_0^T \cos\theta\cdot dt = xT$$ where $\theta$ is a function of $t$ I also have, $$\int_0^T \sin\theta\cdot dt = y$$ I want to solve for $T$. If the ...
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### A solid rocket model: a differential equations set with ending time unknown

I am modelling a rocket model. Consider a solid rocket motor, (let us for sake of simplicity assume that the propellant is distributed in the case with a cylindrical shape: see shape in fig.1 of the ...
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### asymptotics of the solution of an integral equation

Suppose we are given the integral equation $$u(x;a) =v(x)+\int_0^a K(x,y)\,u(y;a)\,dy,$$ where $K(x,y)$ and $v(x)$ are known functions, and $a>0$ is a constant. What I am interested in is the ...