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The original is slove $\dfrac {df} {dt}+\dfrac {t} {1+t}f=at(ie.a*t)$, IC:$f\left( 0\right) =f_{0}$

I used method of variation of constant and get a indefinite integral $\int \dfrac {et} {1+t}dt$,

it is similar to the indefinite integral $\int \dfrac {e^t} {t}dt$, I use integration by parts and get a infinite series

$e^{t}(t^{-1}-t^{-2}+\ldots +\left( n-1\right) !\left( -1\right) ^{n+1}t^{-n}+\ldots)$,

then I got troubled. I don't know what i should do next.

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is that $e^t$ ?? – Santosh Linkha Feb 28 '13 at 15:37
Your equation is incomplete.It has nothing on the other end of= Unless that is a*t and not at. – Ishan Banerjee Feb 28 '13 at 15:37
@experimentX Yes – Jebei Feb 28 '13 at 15:39
Write $1+t = u$ you get $t = u - 1$ so $e^t = e^u/e$, put the $e$ outside, and you get $\int \frac{e^u}{u} du$, unfortunately the integral is non elementary. Check this out ... however with bounds, you can evaluate it numerically. – Santosh Linkha Feb 28 '13 at 15:48
talyor expansion? – user39843 Feb 28 '13 at 15:49
up vote 1 down vote accepted

$\int \dfrac{e^t}{1+t}dt=\int \dfrac{e^v}{ev}dv=\dfrac{Ei(v)}{e}+C=\dfrac{Ei(t+1)}{e}+C$ For properties of this function see,

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