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View Equation

The database contains 327 equations (8 equations are awaiting activation).

Equation data
Category:5. Integral Equations
Subcategory:5.6. Nonlinear Equations with Constant Limits of Integration
Equation(s):$\displaystyle y(x)+\int^{\infty}_{-\infty}\bl[\lambda e^{-|x-t|}y(t)+\varphi(x)\psi(t)y^2(t)]\,dt=f(x)$.
Solution(s),
Transformation(s),
Integral(s)
:
Solutions for $\lambda>-\frac12$:
$$
y_{1,2}(x)=Y_f(x)+A_{1,2}Y_\varphi(x),
$$
where
$$
Y_f(x)=f(x)-\frac{\lambda}{\sqrt{1+2\lambda}}\int^{\infty}_{-\infty}
\exp\bl(-\sqrt{1+2\lambda}\,|x-t|\br)f(t)\,dt,
$$
$$
Y_\varphi(x)&=\varphi(x)-\frac{\lambda}{\sqrt{1+2\lambda}}\int^{\infty}_{-\infty}
\exp\bl(-\sqrt{1+2\lambda}\,|x-t|\br)\varphi(t)\,dt,
$$
and $A_{1,2}$ are roots of the quadratic equation
$$
pA^2+qA+r=0,
$$
$$
p=\int^\infty_0\psi(t)Y_\varphi^2(t)\,dt, \ \ \
q=1+2\int^\infty_0\psi(t)Y_f(t)Y_\varphi(t)\,dt, \ \ \
r=\int^\infty_0\psi(t)Y_f^2(t)\,dt.\cr}
$$
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 17:15
Edits by author:0

Edit (Only for author/contributor)


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