67.8.9 problem 13.2 (c)

Internal problem ID [16504]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 13. Higher order equations: Extending first order concepts. Additional exercises page 259
Problem number : 13.2 (c)
Date solved : Thursday, October 02, 2025 at 01:35:41 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 33

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

\begin{align*} y &= 0 \\ y &= \sqrt {2 c_1 x +2 c_2} \\ y &= -\sqrt {2 c_1 x +2 c_2} \\ \end{align*}

✓ Mathematica. Time used: 0.104 (sec). Leaf size: 20

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

\begin{align*} y(x)&\to c_2 \sqrt {2 x-c_1} \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