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Equation data
Category:5. Integral Equations
Subcategory:5.2. Linear Equations of the Second Kind with Variable Limit of Integration
Equation(s):$\displaystyle y(x)+A\int^x_a\sin[\lambda(x-t)]y(t)\,dt=f(x)$.
Solution(s),
Transformation(s),
Integral(s)
:
1. Solution with $\lambda(A+\lambda)>0$:
$$
y(x)=f(x)-\frac{A\lambda}k\int^x_a\sin[k(x-t)]f(t)\,dt,\qquad
\hbox{where}\quad k=\sqrt{\lambda(A+\lambda)}.
$$

2. Solution with $\lambda(A+\lambda)<0$:
$$
y(x)=f(x)-\frac{A\lambda}k\int^x_a\sinh[k(x-t)]f(t)\,dt,\qquad
\hbox{where}\quad k=\sqrt{-\lambda(\lambda+A)}.
$$

3. Solution with $A=-\lambda$:
$$
y(x)=f(x)+\lambda^2\int^x_a(x-t)f(t)\,dt.
$$
Novelty:Material has been fully published elsewhere
References:A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998.
Author/Contributor's Details
Last name:Polyanin
First name:Andrei
Country:Russia
City:Moscow
Statistic information
Submission date:Wed 01 Aug 2007 10:09
Edits by author:0

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