We have the need to solve in-homogeneous Fredholm equations of the second kind:

\[CurlyPhi][t] - \[Lambda] \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(a\), \(b\)]\(K[t, 
      s] \[CurlyPhi][s] \[DifferentialD]s\)\) == f[t]

Generally, such equations don't have exact solutions. Only in the special cases such equations can be exactly solved..

For example, the equation with $K[t,s]=(1+t^2) s^2$, $f[t]=t^3-\lambda (1+t^2) \frac{32}{3}$, $a=0, b=2$ has an exact solution and may be solved easily analytically.

In the general case, such equations are difficult to solve.

One approach is to reduce them to differential equations. But we are interested in the direct methods without reductions to differential equations.

Does Mathematicat has some tools to solve directly such equations?