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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):$\displaystyle 
y(x)-\int^\infty_0[\lambda\sin(xt)+\varphi(x)\Psi(t,y(t))]y(t)\,dt=f(x)$.
Solution(s),
Transformation(s),
Integral(s)
:
Solutions for $\lambda\ne\pm\sqrt{\frac 2\pi}$:
$$
y_m(x)=Y_f(x)+A_mY_\varphi(x),
$$
where
$$
Y_f(x)&=\frac{f(x)}{1-\frac\pi2\lambda^2}+
\frac{\lambda}{1-\frac\pi2\lambda^2}\int^{\infty}_0\sin(xt)f(t)\,dt,
$$
$$
Y_\varphi(x)&=\frac{\varphi(x)}{1-\frac\pi2\lambda^2}+
\frac{\lambda}{1-\frac\pi2\lambda^2}\int^{\infty}_0\sin(xt)\varphi(t)\,dt,
$$
and $A_m$ are roots of the
algebraic (transcendental) equation
$$
A-\int^b_a \Psi(t,Y_f(t)+AY_\varphi(t))\,dt=0.
$$
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 17:20
Edits by author:0

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