Solving integral equation

Question

Solve the following integral equation:

$${u(x)}={x}+{e^{x}}+\int_{0}^{1}\left({5}{x}^{2}{t}^{2} -{3}{t}^{2}+{t}\right) {u(t)}dt. $$

Answer 1

Write the equation in the form $$u(x)=x + e^x + x^2 \int_0^1 5 t^2 u(t)dt + \int_0^1 (t-3t^2)u(t)dt$$ and denote $$a := \int_0^1 5 t^2 u(t)dt, \quad b := \int_0^1 (t-3t^2)u(t)dt.$$ Then $u(x)=x + e^x + a x^2 + b$. Now find $a$ and $b$ by substituting the last expression (for $u$) in the above formulas for $a$ and $b$.