75.11.7 problem 266
Internal
problem
ID
[16787]
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
:
Monday, March 31, 2025 at 03:17:50 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} y \left (y-2 x y^{\prime }\right )^{2}&=2 y^{\prime } \end{align*}
✓ Maple. Time used: 0.550 (sec). Leaf size: 99
ode:=y(x)*(y(x)-2*x*diff(y(x),x))^2 = 2*diff(y(x),x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {1}{2 \sqrt {-x}} \\
y &= \frac {1}{2 \sqrt {-x}} \\
y &= 0 \\
y &= \frac {\sqrt {x \left (c_1 +x \right )}}{c_1 \sqrt {x}} \\
y &= \frac {\sqrt {x \left (-c_1 +x \right )}}{c_1 \sqrt {x}} \\
y &= -\frac {\sqrt {x \left (c_1 +x \right )}}{c_1 \sqrt {x}} \\
y &= -\frac {\sqrt {x \left (-c_1 +x \right )}}{c_1 \sqrt {x}} \\
\end{align*}
✓ Mathematica. Time used: 3.108 (sec). Leaf size: 158
ode=y[x]*(y[x]-2*x*D[y[x],x])^2==2*D[y[x],x];
ic={};
DSolve[{ode,ic},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*}
✓ Sympy. Time used: 172.186 (sec). Leaf size: 279
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((-2*x*Derivative(y(x), x) + y(x))**2*y(x) - 2*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {\sqrt {C_{1} \left (C_{1} x - 2\right )}}{2}, \ y{\left (x \right )} = \frac {\sqrt {C_{1} \left (C_{1} x - 2\right )}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {C_{1} \left (C_{1} x + 2\right )}}{2}, \ y{\left (x \right )} = \frac {\sqrt {C_{1} \left (C_{1} x + 2\right )}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {- C_{1} x - 2 \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {- C_{1} x - 2 \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {- C_{1} x + 2 \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {- C_{1} x + 2 \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {C_{1} \left (C_{1} x - 2\right )}}{2}, \ y{\left (x \right )} = \frac {\sqrt {C_{1} \left (C_{1} x - 2\right )}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {C_{1} \left (C_{1} x + 2\right )}}{2}, \ y{\left (x \right )} = \frac {\sqrt {C_{1} \left (C_{1} x + 2\right )}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {- C_{1} x - 2 \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {- C_{1} x - 2 \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {- C_{1} x + 2 \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {- C_{1} x + 2 \sqrt {- C_{1}}}}{2}\right ]
\]