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How do I solve the following Volterra, non-homogeneous, $1st$ kind Integral Equation :

$$ \dfrac{x^2}{2}=\int_0^x (1-x^2+t^2)u(t) dt$$

I know I cannot apply Laplace Transform because the kernel is not a "difference kernel". I tried method of successive approximations, but they do not converge.

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Two successive derivations leads to a first order linear ODE :

enter image description here

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