87.1.21 problem 26

Internal problem ID [23233]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 1. Elementary methods. First order differential equations. Exercise at page 9
Problem number : 26
Date solved : Thursday, October 02, 2025 at 09:24:41 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y^{\prime \prime }&=y^{\prime } \left (y^{\prime }+y\right ) \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 30

ode:=diff(diff(y(x),x),x) = diff(y(x),x)*(diff(y(x),x)+y(x)); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.2 (sec). Leaf size: 102

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

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

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(y(x)**2 + 4*Derivative(y(x), (x, 2)))/2 + y(x)/2 + Derivat