How to solve Fredholm integral equation of the second kind? ($f(x) - \lambda\int\limits_0^1 9xtf(t) dt = ax^2 - 4x^2$)

Question

I have an equation : $\ f(x) - \lambda\int\limits_0^1 9xtf(t) dt = ax^2 - 4x^2\ $ in $\ L_2[0,1]\ $ space.

And I want to understand how to solve it, not just obtain an answer.

Answer 1

For this particular equation, it is simple. Differentiate both sides of the equation with respect to $x$: $$ f^\prime(x) = 2 (a-4) x + \lambda \int_0^1 9 t f(t) \mathrm{d} t $$ Since the integral is independent of $x$, the solution is a quadratic $f(x) = (a-4) x^2 + b x + c$. Substituting into the original equation we get a linear system of equations for $b$ and $c$: $$ (a-4) x^2 + b x + c - \lambda x \int_0^1 9 t f(t) \mathrm{d} t = (a-4) x^2 $$ Since $\int_0^1 9 f(t) \mathrm{d} t = \frac{9}{4} a+ 3b+ \frac{9}{2} c - 9$, we get the system: $$ c = 0 \qquad b (1-3 \lambda) - \frac{9}{4} \lambda \left( 2 c + a -4\right) = 0 $$ Therefore we obtain: $$ f(x) = (a-4) x \left( x -\frac{9}{4} \frac{\lambda}{3 \lambda - 1} \right) $$