71.1.86 problem 113 (page 172)

Internal problem ID [19262]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 113 (page 172)
Date solved : Thursday, October 02, 2025 at 04:03:46 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y y^{\prime \prime }+{y^{\prime }}^{2}&=\ln \left (y\right ) y^{2} \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: 1.715 (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: _X0**2 < 2