82.8.9 problem 36-10

Internal problem ID [21880]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 36. Nonlinear differential equations. Page 1203
Problem number : 36-10
Date solved : Thursday, October 02, 2025 at 08:03:18 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 17

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

\[ y = \frac {\sec \left (x \right )}{c_1 \tan \left (x \right )-c_2} \]

✓ Mathematica. Time used: 0.109 (sec). Leaf size: 13

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

\begin{align*} y(x)&\to c_2 \sec (x+c_1) \end{align*}

✗ Sympy

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

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