69.14.20 problem 346

Internal problem ID [18221]
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 : 346
Date solved : Thursday, October 02, 2025 at 03:09:23 PM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.055 (sec). Leaf size: 21

ode:=diff(diff(y(x),x),x) = diff(y(x),x)*ln(diff(y(x),x)); 
ic:=[y(0) = 0, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -\operatorname {Ei}_{1}\left (-2 i {\mathrm e}^{x} \pi \_Z1 \right )+\operatorname {Ei}_{1}\left (-2 i \pi \_Z1 \right ) \]

✗ Mathematica

ode=D[y[x],{x,2}]==D[y[x],x]*Log[D[y[x],x]]; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

{}

✗ Sympy

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

ValueError : Couldnt solve for initial conditions