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integral equation
An integral equation involves an unknown function under the integral sign. Most common of them is a linear integral equation
$\displaystyle\alpha(t)\,y(t)+\!\int_{a}^{b}k(t,\,x)\,y(x)\,dx=f(t),$  (1) 
where $\alpha,\,k,\,f$ are given functions. The function $t\mapsto y(t)$ is to be solved.
Any linear integral equation is equivalent to a linear differential equation; e.g. the equation $\displaystyle y(t)\!+\!\int_{0}^{t}(2t2x3)\,y(x)\,dx=1+t4\sin{t}$ to the equation $y^{{\prime\prime}}(t)3y^{{\prime}}(t)+2y(t)=4\sin{t}$ with the initial conditions $y(0)=1$ and $y^{{\prime}}(0)=0$.
The equation (1) is of

1st kind if $\alpha(t)\equiv 0$,

2nd kind if $\alpha(t)$ is a nonzero constant,

3rd kind else.
If both limits of integration in (1) are constant, (1) is a Fredholm equation, if one limit is variable, one has a Volterra equation. In the case that $f(t)\equiv 0$, the linear integral equation is homogeneous.
Example. Solve the Volterra equation $\displaystyle y(t)\!+\!\int_{0}^{t}(t\!\!x)\,y(x)\,dx=1$ by using Laplace transform.
Using the convolution, the equation may be written $y(t)+t*y(t)=1$. Applying to this the Laplace transform, one obtains $\displaystyle Y(s)+\frac{1}{s^{2}}Y(s)=\frac{1}{s}$, whence $\displaystyle Y(s)=\frac{s}{s^{2}+1}$. This corresponds the function $y(t)=\cos{t}$, which is the solution.
Solutions on some integral equations in EqWorld.
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