= Solution
The given particular solution $y_p=x^2$ suggests the standard reduction of a <Riccati equation>. Substitution of $y=x^2+1/v$ gives
$$
-2x^3v'=5x^2v+2,
\qquad v'+\frac{5}{2x}v=-\frac1{x^3}.
$$
On either side of $x=0$, an <integrating factor> is $|x|^{5/2}$, so $v=(C-2\sqrt{|x|})/|x|^{5/2}$. Hence
$$
\boxed{y=x^2\frac{C-\sqrt{|x|}}{C-2\sqrt{|x|}},}
$$
along with the separately recovered particular solution $y=x^2$. The denominator must remain nonzero on the solution interval; $x=0$ is a singular point of the divided differential equation.
For the positive-$x$ branch, $y(1)=(C-1)/(C-2)$ and $y(4)=16(C-2)/(C-4)$. Nonsingular endpoint values exclude $C=2,4$, so
$$
\boxed{y(4)\bigl(3-2y(1)\bigr)=16.}
$$
The particular solution $y=x^2$ gives the same value directly. These evaluations concern one solution branch, with a common integration constant, rather than independently chosen solutions at the two points.
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