Obtain the solution of the equation$y(x)=F(x)+\lambda \int_a^b K(x,\xi)y(\xi)d\xi $ Where,$K(x,\xi)=u(x)v(\xi)$.

Question

Obtain the solution of the equation$$y(x)=F(x)+\lambda \int_a^b K(x,\xi)y(\xi)d\xi $$ Where,$$K(x,\xi)=u(x)v(\xi)$$

in the form $$y(x)=F(x)+\lambda \frac{\int_a^b K(x,\xi)F(\xi)d\xi }{1-\lambda\int_a^b K(x,x)dx }$$

Solution:We're given that $$y(x)=F(x)+\lambda \int_a^b K(x,\xi)y(\xi)d\xi \tag{$*$}$$ Where,$$K(x,\xi)=u(x)v(\xi)$$ Putting the value of $K(x,\xi)$ in the equation (*),we get $$y(x)=F(x)+\lambda \int_a^b u(x)v(\xi)y(\xi)d\xi=F(x)+\lambda u(x)\int_a^b v(\xi)y(\xi)d\xi$$.

Let,$c=\int_a^b v(\xi)y(\xi)d\xi$.

Then,$y(x)=F(x)+\lambda u(x)c$

Now, i need to determine the value of "$c$".But, i'm not getting, how to determine it?

Please give some hint/suggestions about the method by which i can proceed further...

Thank you...

Answer 1

Multiplying the equation $y(x)=F(x)+\lambda\,c\,u(x)$ by $v(x)$ and integration $[a,b]$ we get $$ c=\int_a^bF(x)\,v(x)\,dx+\lambda\,c\int_a^bu(x)\,v(x)\,dx. $$ This gives you the value of $c$.