Solve integral equation of second kind using Fredholm method

Question

I need to solve this integral equation

$$\phi (x)=(x^2-x^4)+ \lambda \int_{-1}^{1}(x^4+5x^3y)\phi (y)dy$$

Using the Fredholm theory of the intergalactic equations of second kind. I really don't understand the method. Can you please explain this to me so I can solve the other exercises??

Thanks a lot!

Answer 1

$\phi(x)=x^2-x^4+\lambda\int_{-1}^1(x^4+5x^3y)\phi(x)~dy$

$\phi(x)=x^2-x^4+\left[\lambda\left(x^4y+\dfrac{5x^3y^2}{2}\right)\phi(x)\right]_{-1}^1$

$\phi(x)=x^2-x^4+2\lambda x^4\phi(x)$

$(2\lambda x^4-1)\phi(x)=x^4-x^2$

$\phi(x)=\dfrac{x^4-x^2}{2\lambda x^4-1}$

Answer 2

You can use the direct computation method. Here is the final result

$$ \phi \left( x \right) =-{x}^{4}+{x}^{2}-\frac{4}{3}\,{\frac {\lambda\,{x}^{4}} {2\,\lambda-5}}. $$

Note that, $\lambda = \frac{5}{2}$ is a singular value.