Internal problem ID [15128]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Section 11. Singular solutions of differential equations. Exercises page 92
Problem number: 266.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class G`], _rational]
\[ \boxed {y \left (y-2 x y^{\prime }\right )^{2}-2 y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 99
dsolve(y(x)*(y(x)-2*x*diff(y(x),x))^2=2*diff(y(x),x),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {1}{2 \sqrt {-x}} \\ y \left (x \right ) &= \frac {1}{2 \sqrt {-x}} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\sqrt {\left (x +c_{1} \right ) x}}{c_{1} \sqrt {x}} \\ y \left (x \right ) &= \frac {\sqrt {x \left (x -c_{1} \right )}}{c_{1} \sqrt {x}} \\ y \left (x \right ) &= -\frac {\sqrt {\left (x +c_{1} \right ) x}}{c_{1} \sqrt {x}} \\ y \left (x \right ) &= -\frac {\sqrt {x \left (x -c_{1} \right )}}{c_{1} \sqrt {x}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.935 (sec). Leaf size: 158
DSolve[y[x]*(y[x]-2*x*y'[x])^2==2*y'[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {2} \sqrt {e^{-2 c_1} \left (2 x-e^{c_1}\right )} \\ y(x)\to \sqrt {2} \sqrt {e^{-2 c_1} \left (2 x-e^{c_1}\right )} \\ y(x)\to -\sqrt {2} \sqrt {e^{-2 c_1} \left (2 x+e^{c_1}\right )} \\ y(x)\to \sqrt {2} \sqrt {e^{-2 c_1} \left (2 x+e^{c_1}\right )} \\ y(x)\to 0 \\ y(x)\to -\frac {i}{2 \sqrt {x}} \\ y(x)\to \frac {i}{2 \sqrt {x}} \\ \end{align*}