52.1.23 problem 23

Internal problem ID [10394]
Book : Second order enumerated odes
Section : section 1
Problem number : 23
Date solved : Tuesday, September 30, 2025 at 07:23:01 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+{y^{\prime }}^{2}+y&=0 \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 61

ode:=diff(diff(y(x),x),x)+diff(y(x),x)^2+y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} -2 \int _{}^{y}\frac {1}{\sqrt {2+4 \,{\mathrm e}^{-2 \textit {\_a}} c_1 -4 \textit {\_a}}}d \textit {\_a} -x -c_2 &= 0 \\ 2 \int _{}^{y}\frac {1}{\sqrt {2+4 \,{\mathrm e}^{-2 \textit {\_a}} c_1 -4 \textit {\_a}}}d \textit {\_a} -x -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.241 (sec). Leaf size: 308

ode=D[y[x],{x,2}]+(D[y[x],x])^2+y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{K[2]}}{\sqrt {c_1+2 \int _1^{K[2]}-e^{2 K[1]} K[1]dK[1]}}dK[2]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{K[3]}}{\sqrt {c_1+2 \int _1^{K[3]}-e^{2 K[1]} K[1]dK[1]}}dK[3]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{K[2]}}{\sqrt {2 \int _1^{K[2]}-e^{2 K[1]} K[1]dK[1]-c_1}}dK[2]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{K[2]}}{\sqrt {c_1+2 \int _1^{K[2]}-e^{2 K[1]} K[1]dK[1]}}dK[2]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{K[3]}}{\sqrt {2 \int _1^{K[3]}-e^{2 K[1]} K[1]dK[1]-c_1}}dK[3]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{K[3]}}{\sqrt {c_1+2 \int _1^{K[3]}-e^{2 K[1]} K[1]dK[1]}}dK[3]\&\right ][x+c_2] \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) + Derivative(y(x), x)**2 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(-y(x) - Derivative(y(x), (x, 2))) + Derivative(y(x), x) cannot be solved by the factorable group method