89.31.13 problem 13
Internal
problem
ID
[24958]
Book
:
A
short
course
in
Differential
Equations.
Earl
D.
Rainville.
Second
edition.
1958.
Macmillan
Publisher,
NY.
CAT
58-5010
Section
:
Chapter
16.
Equations
of
order
one
and
higher
degree.
Exercises
at
page
246
Problem
number
:
13
Date
solved
:
Thursday, October 02, 2025 at 11:36:37 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} y {y^{\prime }}^{2}-x y^{\prime }+y&=0 \end{align*}
✓ Maple. Time used: 0.046 (sec). Leaf size: 164
ode:=y(x)*diff(y(x),x)^2-x*diff(y(x),x)+y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
\frac {4 \ln \left (\frac {y}{x}\right ) y^{2}+\left (4 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-\frac {4 y^{2}-x^{2}}{x^{2}}}}\right )-4 c_1 +4 \ln \left (x \right )\right ) y^{2}-x^{2} \left (\sqrt {\frac {x^{2}-4 y^{2}}{x^{2}}}-1\right )}{4 y^{2}} &= 0 \\
\frac {4 \ln \left (\frac {y}{x}\right ) y^{2}+\left (-4 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-\frac {4 y^{2}-x^{2}}{x^{2}}}}\right )-4 c_1 +4 \ln \left (x \right )\right ) y^{2}+x^{2} \left (\sqrt {\frac {x^{2}-4 y^{2}}{x^{2}}}+1\right )}{4 y^{2}} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.819 (sec). Leaf size: 232
ode=y[x]*D[y[x],x]^2-x*D[y[x],x]+y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} \text {Solve}\left [\frac {1}{2} \log \left (1+i \sqrt {\frac {2 y(x)}{x}-1} \sqrt {\frac {2 y(x)}{x}+1}\right )-\frac {\sqrt {\frac {2 y(x)}{x}-1} \sqrt {\frac {2 y(x)}{x}+1}}{2 \sqrt {\frac {2 y(x)}{x}-1} \sqrt {\frac {2 y(x)}{x}+1}-2 i}=-\frac {\log (x)}{2}+c_1,y(x)\right ]\\ \text {Solve}\left [\frac {1}{2} \log \left (1-i \sqrt {\frac {2 y(x)}{x}-1} \sqrt {\frac {2 y(x)}{x}+1}\right )-\frac {\sqrt {\frac {2 y(x)}{x}-1} \sqrt {\frac {2 y(x)}{x}+1}}{2 \sqrt {\frac {2 y(x)}{x}-1} \sqrt {\frac {2 y(x)}{x}+1}+2 i}=-\frac {\log (x)}{2}+c_1,y(x)\right ]\\ y(x)&\to 0 \end{align*}
✓ Sympy. Time used: 18.007 (sec). Leaf size: 119
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x*Derivative(y(x), x) + y(x)*Derivative(y(x), x)**2 + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \log {\left (x \right )} = C_{1} - \log {\left (\sqrt {2} \sqrt {\sqrt {1 - \frac {4 y^{2}{\left (x \right )}}{x^{2}}} - 1 + \frac {2 y^{2}{\left (x \right )}}{x^{2}}} \right )} + \frac {1}{\sqrt {1 - \frac {4 y^{2}{\left (x \right )}}{x^{2}}} - 1}, \ \log {\left (x \right )} = C_{1} - \log {\left (\sqrt {2} \sqrt {- \sqrt {1 - \frac {4 y^{2}{\left (x \right )}}{x^{2}}} - 1 + \frac {2 y^{2}{\left (x \right )}}{x^{2}}} \right )} - \frac {1}{\sqrt {1 - \frac {4 y^{2}{\left (x \right )}}{x^{2}}} + 1}\right ]
\]