69.14.22 problem 348

Internal problem ID [18223]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 14. Differential equations admitting of depression of their order. Exercises page 107
Problem number : 348
Date solved : Thursday, October 02, 2025 at 03:09:25 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y^{\prime \prime }&=y^{\prime } \left (y^{\prime }+1\right ) \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 16

ode:=diff(diff(y(x),x),x) = diff(y(x),x)*(1+diff(y(x),x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\ln \left (-c_1 \,{\mathrm e}^{x}-c_2 \right ) \]

✓ Mathematica. Time used: 0.446 (sec). Leaf size: 41

ode=D[y[x],{x,2}]==D[y[x],x]*(1+D[y[x],x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.562 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-Derivative(y(x), x) - 1)*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} - \log {\left (C_{2} + e^{x} \right )} \]