41.2.27 problem 27
Internal
problem
ID
[8729]
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
:
27
Date
solved
:
Tuesday, September 30, 2025 at 05:45:51 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} y {y^{\prime }}^{2}+2 x y^{\prime }-y&=0 \end{align*}
✓ Maple. Time used: 0.174 (sec). Leaf size: 69
ode:=y(x)*diff(y(x),x)^2+2*x*diff(y(x),x)-y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -i x \\
y &= i x \\
y &= 0 \\
y &= \sqrt {c_1 \left (c_1 -2 x \right )} \\
y &= \sqrt {c_1 \left (c_1 +2 x \right )} \\
y &= -\sqrt {c_1 \left (c_1 -2 x \right )} \\
y &= -\sqrt {c_1 \left (c_1 +2 x \right )} \\
\end{align*}
✓ Mathematica. Time used: 0.26 (sec). Leaf size: 126
ode=y[x]*(D[y[x],x])^2+2*x*D[y[x],x]-y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -e^{\frac {c_1}{2}} \sqrt {-2 x+e^{c_1}}\\ y(x)&\to e^{\frac {c_1}{2}} \sqrt {-2 x+e^{c_1}}\\ y(x)&\to -e^{\frac {c_1}{2}} \sqrt {2 x+e^{c_1}}\\ y(x)&\to e^{\frac {c_1}{2}} \sqrt {2 x+e^{c_1}}\\ y(x)&\to 0\\ y(x)&\to -i x\\ y(x)&\to i x \end{align*}
✓ Sympy. Time used: 141.202 (sec). Leaf size: 469
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x*Derivative(y(x), x) + y(x)*Derivative(y(x), x)**2 - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{2}\right ]
\]