73.9.7 problem 14.1 (g)

Internal problem ID [15197]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 14. Higher order equations and the reduction of order method. Additional exercises page 277
Problem number : 14.1 (g)
Date solved : Monday, March 31, 2025 at 01:31:05 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]

\begin{align*} \left (y+1\right ) y^{\prime \prime }&={y^{\prime }}^{3} \end{align*}

✓ Maple. Time used: 0.033 (sec). Leaf size: 40

ode:=(1+y(x))*diff(diff(y(x),x),x) = diff(y(x),x)^3; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -1 \\ y &= c_1 \\ y &= \frac {-c_1 -c_2 -x}{\operatorname {LambertW}\left (-\left (c_1 +c_2 +x \right ) {\mathrm e}^{-c_1 -1}\right )}-1 \\ \end{align*}

✓ Mathematica. Time used: 0.4 (sec). Leaf size: 93

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

\begin{align*} y(x)\to \text {InverseFunction}[\text {$\#$1}-(\text {$\#$1}+1) \log (\text {$\#$1}+1)+\text {$\#$1} (-c_1)\&][x+c_2] \\ y(x)\to \text {InverseFunction}[\text {$\#$1}-(\text {$\#$1}+1) \log (\text {$\#$1}+1)+\text {$\#$1} (-(-c_1))\&][x+c_2] \\ y(x)\to \text {InverseFunction}[\text {$\#$1}-(\text {$\#$1}+1) \log (\text {$\#$1}+1)+\text {$\#$1} (-c_1)\&][x+c_2] \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE (y(x)*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 2)))**(1/3)/2 - sqrt(3)*I*(y(x)*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 2)))**(1/3)/2 + Derivative(y(x), x) cannot be solved by the factorable group method