9.18.38 problem 38

Internal problem ID [3281]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 35, page 157
Problem number : 38
Date solved : Tuesday, September 30, 2025 at 06:32:20 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 2 y y^{\prime \prime }&=y^{3}+2 {y^{\prime }}^{2} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.525 (sec). Leaf size: 15

ode:=2*y(x)*diff(diff(y(x),x),x) = y(x)^3+2*diff(y(x),x)^2; 
ic:=[y(0) = -1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \operatorname {RootOf}\left (2 \,\operatorname {arctanh}\left (\sqrt {\textit {\_Z} +1}\right )+x \right ) \]

✓ Mathematica. Time used: 53.474 (sec). Leaf size: 15

ode=2*y[x]*D[y[x],{x,2}]==y[x]^3+2*D[y[x],x]^2; 
ic={y[0]==-1,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\text {sech}^2\left (\frac {x}{2}\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x)**3 + 2*y(x)*Derivative(y(x), (x, 2)) - 2*Derivative(y(x), x)**2,0) 
ics = {y(0): -1, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

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