72.7.9 problem 2 (b)

Internal problem ID [19481]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 2. First order equations. Section 11 (Reduction of order). Problems at page 87
Problem number : 2 (b)
Date solved : Thursday, October 02, 2025 at 04:31:39 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _with_potential_symmetries], [_2nd_order, _reducible, _mu_xy]]

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

With initial conditions

\begin{align*} y \left (0\right )&=-{\frac {1}{2}} \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.221 (sec). Leaf size: 16

ode:=y(x)*diff(diff(y(x),x),x) = y(x)^2*diff(y(x),x)+diff(y(x),x)^2; 
ic:=[y(0) = -1/2, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -\frac {3}{8 \,{\mathrm e}^{\frac {3 x}{2}}-2} \]

✓ Mathematica. Time used: 1.617 (sec). Leaf size: 20

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

\begin{align*} y(x)&\to \frac {3}{2-8 e^{3 x/2}} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -sqrt((y(x)**3 + 4*Derivative(y(x), (x, 2)))*y(x))/2 + y(x)**2/2 + Derivative(y(x), x) cannot be solved by the factorable group method