= Solution
{c}
With the standard constant starting <Picard iterate>, $y_0(x)=1$, the next two <Picard iterates> are
$$
y_1(x)=1+\int_1^x\left(s+\frac{s}{1}\right)\,ds=x^2
$$
and, for $x>0$,
$$
\begin{aligned}
y_2(x)&=1+\int_1^x\left(s+\frac{s}{s^2}\right)\,ds\\
&=1+\frac{x^2-1}{2}+\log x.
\end{aligned}
$$
Thus
$$
\boxed{y_0(x)=1,\qquad y_1(x)=x^2,\qquad y_2(x)=\frac{x^2+1}{2}+\log x.}
$$
The restriction $x>0$ for $y_2$ is essential for the <integral> from $1$ to $x$: the preceding iterate $y_1$ vanishes at $0$, where the new integrand $s+1/s$ is undefined.
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