= Solution
On the component $x>0$ containing the boundary point, divide by $2x$ and use <integrating factor> $x^{-1/2}$:
$$
\frac d{dx}(x^{-1/2}y)=\frac12x^{3/2}.
$$
Integration gives $y=x^3/5+C\sqrt x$, and $y(1)=1$ fixes $C=4/5$. Therefore
$$
\boxed{y(x)=\frac{x^3+4\sqrt x}{5},\qquad x>0.}
$$
Although the function has a continuous limit at zero, its derivative diverges there; the solution with this boundary datum does not extend as a differentiable solution through the singular point of the equation.
Back to article page
