Internal problem ID [2820]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 3
Problem number: 65.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-x \left (2+y x^{2}-y^{2}\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 67
dsolve(diff(y(x),x) = x*(2+x^2*y(x)-y(x)^2),y(x), singsol=all)
\[ y \relax (x ) = \frac {2 c_{1} {\mathrm e}^{-\frac {x^{4}}{4}}}{\sqrt {\pi }\, \left (\erf \left (\frac {x^{2}}{2}\right ) c_{1}+1\right )}+\frac {\erf \left (\frac {x^{2}}{2}\right ) \sqrt {\pi }\, c_{1} x^{2}+x^{2} \sqrt {\pi }}{\sqrt {\pi }\, \left (\erf \left (\frac {x^{2}}{2}\right ) c_{1}+1\right )} \]
✓ Solution by Mathematica
Time used: 0.255 (sec). Leaf size: 48
DSolve[y'[x]==x(2+x^2 y[x]-y[x]^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x^2+\frac {2 e^{-\frac {x^4}{4}}}{\sqrt {\pi } \text {Erf}\left (\frac {x^2}{2}\right )+2 c_1} \\ y(x)\to x^2 \\ \end{align*}