75.11.7 problem 266

Internal problem ID [16858]
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
Date solved : Tuesday, January 28, 2025 at 09:35:29 AM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} y \left (y-2 x y^{\prime }\right )^{2}&=2 y^{\prime } \end{align*}

✓ Solution by Maple

Time used: 0.327 (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 &= -\frac {1}{2 \sqrt {-x}} \\ y &= \frac {1}{2 \sqrt {-x}} \\ y &= 0 \\ y &= \frac {\sqrt {\left (x +c_{1} \right ) x}}{c_{1} \sqrt {x}} \\ y &= \frac {\sqrt {\left (x -c_{1} \right ) x}}{c_{1} \sqrt {x}} \\ y &= -\frac {\sqrt {\left (x +c_{1} \right ) x}}{c_{1} \sqrt {x}} \\ y &= -\frac {\sqrt {\left (x -c_{1} \right ) x}}{c_{1} \sqrt {x}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 3.108 (sec). Leaf size: 158

DSolve[y[x]*(y[x]-2*x*D[y[x],x])^2==2*D[y[x],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*}