15.18.35 problem 35

Internal problem ID [3278]
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 : 35
Date solved : Sunday, March 30, 2025 at 01:26:18 AM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

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

✓ Maple. Time used: 27.950 (sec). Leaf size: 25

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

\[ y = -\operatorname {sech}\left (\frac {\sqrt {2}\, \left (x -\sqrt {2}\, \operatorname {arctanh}\left (\sqrt {2}\right )\right )}{2}\right )^{2} \]

✗ Mathematica

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

{}

✗ Sympy

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

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