 Solution

The given particular solution $y_p=x^2$ suggests the standard reduction of a Riccati equation. Substitution of $y=x^2+1/v$ givesOn either side of $x=0$, an integrating factor is $|x{|}^{5/2}$, so $v=(C-2\sqrt{|x|})/|x{|}^{5/2}$. Hencealong 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 $C=2,4$, soThe 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.