= Solution
{c}
The <integrating factor> for this <linear ordinary differential equation> is $\mu(x)=e^{x^3/3}$. Multiplication gives an exact <derivative>:
$$
\frac{d}{dx}\left(e^{x^3/3}y\right)=x^2e^{x^3/3}=\frac{d}{dx}e^{x^3/3}.
$$
Consequently $y=1+Ce^{-x^3/3}$. The <initial condition> $y(1)=2$ gives $C=e^{1/3}$, so the <initial value problem> has solution
$$
\boxed{y(x)=1+\exp\left(\frac{1-x^3}{3}\right),\qquad x\in\mathbb R.}
$$
The <integrating factor> calculation determines every solution, and the <initial condition> fixes its remaining <constant of integration> uniquely.
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