# Integral equations

• Nov 17th 2012, 05:51 PM
MAX09
Integral equations
When does the integral equation given by

$\displaystyle y(x) = 1 + \lambda \int_{0}^{\frac{\pi} {2}} {cos (x-t) y(t) dt$ have

i) a unique solution ???
ii) no solution???

• Nov 17th 2012, 09:44 PM
chiro
Re: Integral equations
Hey MAX09.

Have you tried differentiating the integral in order to get a differential equation to solve?

Differentiation under the integral sign - Wikipedia, the free encyclopedia

Also Lipschitz continuity is a good way to show existence of differential equations.

Lipschitz continuity - Wikipedia, the free encyclopedia
• Nov 19th 2012, 07:14 PM
MAX09
Re: Integral equations
Chiro, Thanks for your suggestion. I tried along those lines, but it seems I need specific literature on Linear Integral Equations.
A hint or a link in that direction will definitely give me a headstart.
• Nov 20th 2012, 12:32 AM
JJacquelin
Re: Integral equations
in the integral cos(x-t) = cos(x)cos(t)+sin(x)sin(t)
Leading to y = 1+lambda *(A*cos(x)+B*sin(x))
where A and B are constant (definite integrals where there is no x into them)
Bringing back y = 1+lambda *(A*cos(x)+B*sin(x)) into the equation, after integration we obtain two relationships so that the coefficient of sin(x) and cos(x) are = 0.
This leads to : A = B = -4 / (pi+2-(4/lambda))
Conclusion : There is one solution.