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View Equation

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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):$\displaystyle y(x)+\int^{\infty}_{-\infty}\bl[\lambda e^{-|x-t|}+\varphi(x)\psi(t)]y(t)\,dt=f(x)$.
Solution(s),
Transformation(s),
Integral(s)
:
Solution for $\lambda>-\frac12$:
$$
y(x)=Y_f(x)+AY_\varphi(x),
$$
where
$$
Y_f(x)=f(x)-\frac{\lambda}{\sqrt{1+2\lambda}}\int^{\infty}_{-\infty}
\exp\bl(-\sqrt{1+2\lambda}\,|x-t|\br)f(t)\,dt,
$$
$$
Y_\varphi(x)=\varphi(x)-\frac{\lambda}{\sqrt{1+2\lambda}}\int^{\infty}_{-\infty}
\exp\bl(-\sqrt{1+2\lambda}\,|x-t|\br)\varphi(t)\,dt,
$$
$$
A=-\frac{\int^\infty_0\psi(t)Y_f(t)\,dt}{1+\int^\infty_0\psi(t)Y_\varphi(t)\,dt}.
$$
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 16:10
Edits by author:0

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