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Equation data
Category:5. Integral Equations
Subcategory:5.4. Linear Equations of the Second Kind with Constant Limits of Integration
Equation(s):$\displaystyle y(x)-\int^\infty_0[\lambda\cos(xt)+\varphi(x)\psi(t)]y(t)\,dt=f(x)$.
Solution(s),
Transformation(s),
Integral(s)
:
Solution for $\lambda\ne\pm\sqrt{\frac 2\pi}$:
$$
y(x)=Y_f(x)+AY_\varphi(x),
$$
where
$$
Y_f(x)=\frac{f(x)}{1-\frac\pi2\lambda^2}+
\frac{\lambda}{1-\frac\pi2\lambda^2}\int^{\infty}_0\cos(xt)f(t)\,dt,
$$
$$
Y_\varphi(x)=\frac{\varphi(x)}{1-\frac\pi2\lambda^2}+
\frac{\lambda}{1-\frac\pi2\lambda^2}\int^{\infty}_0\cos(xt)\varphi(t)\,dt,
$$
$$
A=\frac{\int^\infty_0\psi(t)Y_f(t)\,dt}{1-\int^\infty_0\psi(t)Y_\varphi(t)\,dt}.
$$
Here, all integrals are supposed to converge.
Novelty:New equation(s) & solution(s) / integral(s)
Author/Contributor's Details
Last name:Polyanin
First name:Andrei
Country:Russia
City:Moscow
Statistic information
Submission date:Tue 03 Jul 2007 16:18
Edits by author:0

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