Internal problem ID [4997]
Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold
Scientific. Singapore. 1995
Section: Chapter 1. First order differential equations. Section 1.2 Homogeneous equations problems.
page 12
Problem number: 3.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Bernoulli]
\[ \boxed {2 y^{\prime } x -y \left (2 x^{2}-y^{2}\right )=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 74
dsolve(2*x*diff(y(x),x)=y(x)*(2*x^2-y(x)^2),y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {\sqrt {-2 \left (-2 c_{1} +\operatorname {Ei}_{1}\left (-x^{2}\right )\right ) {\mathrm e}^{x^{2}}}}{-2 c_{1} +\operatorname {Ei}_{1}\left (-x^{2}\right )} \\ y \left (x \right ) = -\frac {\sqrt {-2 \left (-2 c_{1} +\operatorname {Ei}_{1}\left (-x^{2}\right )\right ) {\mathrm e}^{x^{2}}}}{-2 c_{1} +\operatorname {Ei}_{1}\left (-x^{2}\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.261 (sec). Leaf size: 65
DSolve[2*x*y'[x]==y[x]*(2*x^2-y[x]^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {e^{\frac {x^2}{2}}}{\sqrt {\frac {\operatorname {ExpIntegralEi}\left (x^2\right )}{2}+c_1}} \\ y(x)\to \frac {e^{\frac {x^2}{2}}}{\sqrt {\frac {\operatorname {ExpIntegralEi}\left (x^2\right )}{2}+c_1}} \\ y(x)\to 0 \\ \end{align*}