Property of a solution of $t^2y'+y=t^2$ I'm currently studying the ODE $t^2y'+y=t^2$.
The general solution on $\left]0,+\infty\right[$ is
$$y=e^{1/t}\left[\int_0^t e^{-1/u}{\rm d}u+C\right]$$
I'm interested in the only solution $f$ having a finite limit at $0^+$, which is given by choosing $C=0$. So
$$f(t)=e^{1/t}\int_0^t e^{-1/u}{\rm d}u$$
Now I know how to find asymptotic development of $f$ at $0$, by successive integrations by parts :
$$f(t) = \sum_{k=2}^n (-1)^k(k-1)!t^k + (-1)^{n+1}n!I_{n-1}(t)e^{1/t}$$
where
$$I_n(t) = \int_0^t u^ne^{-1/u}{\rm d}u = o(t^n)$$
proven by another IPP.
BTW, the only normal series solution of the differential equation is
$$S=\sum_{n=2}^\infty (-1)^n(n-1)!X^n$$
What I want to prove is that by extending $f$ to $0$ with $f(0)=0$, I get a indefinitely differentiable function, the Taylor expansion of which is exactly $S$. The second part seems easy, but I can't find a basic reason which explains why $f$ is $\mathscr C^\infty$.
Can you help me on this ?
Thanks for reading all this ;-)
 A: You alsmost wrote it
$$t^2y'+y=0 \quad \implies \quad y=C\, e^{\frac{1}{t}}$$ Variation of parameter leads to
$$t^2 \left(e^{\frac{1}{t}} C'-1\right)=0\quad \implies \quad C=c_1+\text{Ei}\left(-\frac{1}{t}\right)+t\,e^{-\frac{1}{t}} $$
$$y=c_1\, e^{\frac{1}{t}}+e^{\frac{1}{t}} \,\text{Ei}\left(-\frac{1}{t}\right)+t$$ So, you are concerned by the particular solution
$$y=t+e^{\frac{1}{t}} \,\text{Ei}\left(-\frac{1}{t}\right)=\sum_{n=2}^\infty (-1)^n \,(n-1)!\,t^n$$ where $\text{Ei}$ is the exponentila integral function.
For "large" values of $t$, its asymptotics is given by
$$y=t+(\gamma -\log (t))+\frac{(\gamma -\log (t)) -1}{t}+\frac{2(\gamma -\log (t)) -3}{4
   t^2}+O\left(\frac{1}{t^3}\right)$$ which gives a relative error smaller than $1$% as soon as $t>3$ and smaller than $0.1$% as soon as $t>5$.
Edit
If you wish a long expansion
$$y=t+\sum_{n=0}^\infty\frac{ [\gamma-\log(t)] -a_n}{{n!}\,t^n} $$ and the $a_n$ form the sequence
$$\left\{0,1,\frac{3}{2},\frac{11}{6},\frac{25}{12},\frac{137}{60},\frac{49}{20},\frac{36
   3}{140},\frac{761}{280},\frac{7129}{2520},\frac{7381}{2520},\frac{83711}{27720},\frac{86021}{27720},\cdots\right\}$$
The denominators correspond to sequence $A231693$ in $OEIS$ but the numerators have not been identified.
