When $y(t) = \lambda x(t) - \int_{-1}^{1} (ts^2 + t^2 s^3 + t^3 s) x(s) ds$ has a solution? The question is as follows:
Estabilish conditions for the integral equation
$$y(t) = \lambda x(t) - \int_{-1}^{1} (ts^2 + t^2 s^3 + t^3 s) x(s) ds$$
to have a solution.
$\textbf{Definitions and efforts:}$
Actually I do not know what to do! It's my first of this kind and I am trying to learn!
I just know that it is an integral equation which is said is Fredholm equation because the integration limits $a$ and $b$ are constants. 
Can someone please let me know how can we attack to this problem and what are techniques?
Thanks!
 A: $$y(t) = \lambda x(t) - t\int_{-1}^{1} s^2 x(s) ds
- t^2\int_{-1}^{1} s^3 x(s) ds - t^3\int_{-1}^{1} s x(s) ds$$
$\int_{-1}^{1} s^2 x(s) ds=c_1 \quad;\quad \int_{-1}^{1} s^3 x(s) ds=c_2 \quad;\quad \int_{-1}^{1} s x(s) ds$
$$y(t) = \lambda x(t) - c_1t-c_2t^2-c_3t^3$$
$$x(t)=\frac{1}{\lambda}(y(t)+ c_1t+c_2t^2+c_3t^3)$$
$\int_{-1}^{1} s^2 x(s) ds=c_1= \int_{-1}^{1} s^2\left(\frac{1}{\lambda}(y(s)+ c_1s+c_2s^2+c_3s^3)\right) ds $
$c_1= \frac{1}{\lambda}\int_{-1}^{1} s^2y(s)ds+c_1\frac{1}{\lambda}\int_{-1}^{1}s^3ds +c_2\frac{1}{\lambda}\int_{-1}^{1}s^4ds +c_3\frac{1}{\lambda}\int_{-1}^{1}s^5ds = \frac{1}{\lambda}\int_{-1}^{1} s^2y(s)ds+\frac{2}{5\lambda}c_2$
$c_2= \int_{-1}^{1} s^3\left(\frac{1}{\lambda}(y(s)+ c_1s+c_2s^2+c_3s^3)\right) ds =\frac{1}{\lambda}\int_{-1}^{1} s^3y(s)ds+\frac{2}{5\lambda}c_1+\frac{2}{7\lambda}c_3$
$c_3= \int_{-1}^{1} s\left(\frac{1}{\lambda}(y(s)+ c_1s+c_2s^2+c_3s^3)\right) ds =\frac{1}{\lambda}\int_{-1}^{1} sy(s)ds+\frac{2}{3\lambda}c_1+\frac{2}{5\lambda}c_3$
$\begin{cases}
\lambda c_1-\frac{2}{5}c_2= \int_{-1}^{1} s^2y(s)ds\\
\lambda c_2-\frac{2}{5}c_1-\frac{2}{7}c_3=\int_{-1}^{1} s^3y(s)ds\\
\lambda c_3-\frac{2}{3}c_1-\frac{2}{5}c_3=\int_{-1}^{1} sy(s)ds
\end{cases}$
$$\begin{bmatrix}c_1 \\ c_2 \\c_3\end{bmatrix} =
\begin{bmatrix}
       \lambda & -\frac{2}{5} & 0           \\
       -\frac{2}{5} & \lambda           & -\frac{2}{7} \\
       -\frac{2}{3}           & 0 & \lambda-\frac{2}{5}
     \end{bmatrix}^{-1}
\begin{bmatrix}\int_{-1}^{1} s^2y(s)ds \\ 
\int_{-1}^{1} s^3y(s)ds \\
\int_{-1}^{1} sy(s)ds\end{bmatrix}$$
$$x(t)=\frac{1}{\lambda}\left(y(t)+c_1 t+c_2 t^2+c_3 t^3 \right)$$
