77.1.86 problem 113 (page 172)
Internal
problem
ID
[17905]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
113
(page
172)
Date
solved
:
Monday, March 31, 2025 at 04:49:42 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} y y^{\prime \prime }+{y^{\prime }}^{2}&=y^{2} \ln \left (y\right ) \end{align*}
✓ Maple. Time used: 0.010 (sec). Leaf size: 73
ode:=y(x)*diff(diff(y(x),x),x)+diff(y(x),x)^2 = y(x)^2*ln(y(x));
dsolve(ode,y(x), singsol=all);
\begin{align*}
-4 \int _{}^{y}\frac {\textit {\_a}}{\sqrt {8 \textit {\_a}^{4} \ln \left (\textit {\_a} \right )-2 \textit {\_a}^{4}+16 c_1}}d \textit {\_a} -x -c_2 &= 0 \\
4 \int _{}^{y}\frac {\textit {\_a}}{\sqrt {8 \textit {\_a}^{4} \ln \left (\textit {\_a} \right )-2 \textit {\_a}^{4}+16 c_1}}d \textit {\_a} -x -c_2 &= 0 \\
\end{align*}
✓ Mathematica. Time used: 1.627 (sec). Leaf size: 311
ode=y[x]*D[y[x],{x,2}]+D[y[x],x]^2==y[x]^2*Log[y[x]];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {2 \sqrt {2} K[1]}{\sqrt {4 \log (K[1]) K[1]^4-K[1]^4+8 c_1}}dK[1]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {2 \sqrt {2} K[2]}{\sqrt {4 \log (K[2]) K[2]^4-K[2]^4+8 c_1}}dK[2]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {2 \sqrt {2} K[1]}{\sqrt {4 \log (K[1]) K[1]^4-K[1]^4+8 (-1) c_1}}dK[1]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {2 \sqrt {2} K[1]}{\sqrt {4 \log (K[1]) K[1]^4-K[1]^4+8 c_1}}dK[1]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {2 \sqrt {2} K[2]}{\sqrt {4 \log (K[2]) K[2]^4-K[2]^4+8 (-1) c_1}}dK[2]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {2 \sqrt {2} K[2]}{\sqrt {4 \log (K[2]) K[2]^4-K[2]^4+8 c_1}}dK[2]\&\right ][x+c_2] \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-y(x)**2*log(y(x)) + y(x)*Derivative(y(x), (x, 2)) + Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
TypeError : cannot determine truth value of Relational