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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):$\displaystyle \int^\infty_0[\cos(xt)y(t)+\varphi(x)\psi(t)y^2(t)]\,dt=f(x)$.
Solution(s),
Transformation(s),
Integral(s)
:
Solutions:
$$
y_{1,2}(t)=Y_f(t)+A_{1,2}Y_\varphi(t),
$$
where
$$
Y_f(t)=\frac 2{\pi}\int^{\infty}_0\cos(xt)f(x)\,dx,\quad \
Y_\varphi(t)=\frac 2{\pi}\int^{\infty}_0\cos(xt)\varphi(x)\,dx,
$$
and $A_{1,2}$ are roots of the quadratic equation
$$\displaylines{
pA^2+qA+r=0,\cr
p=\int^\infty_0\psi(t)Y_\varphi^2(t)\,dt, \ \ \
q=1+2\int^\infty_0\psi(t)Y_f(t)Y_\varphi(t)\,dt, \ \ \
r=\int^\infty_0\psi(t)Y_f^2(t)\,dt.\cr}
$$
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 17:05
Edits by author:0

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