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Differential equation for $ F_n$

We have

$\displaystyle I_n(x) = F_n(x) {\rm e}^x - (-1)^n n!$

Differentiating both sides of the equation gives

$\displaystyle x^n {\rm e}^x = F_n'(x) {\rm e}^x + F_n(x) {\rm e}^x

Hence, $ F_n(x)$ satisfies

$\displaystyle y' + y = x^n$ (3)

We will use this to deduce an expression for $ F_n(x)$.

Alexander Frolkin 2001-06-02