26.10.2 problem Exercise 35.2, page 504

Internal problem ID [7131]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 8. Special second order equations. Lesson 35. Independent variable x absent
Problem number : Exercise 35.2, page 504
Date solved : Tuesday, September 30, 2025 at 04:21:52 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.221 (sec). Leaf size: 46

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

\begin{align*} y &= \frac {\sqrt {c_1 \left (\left (c_2 +x \right )^{2} c_1^{2}+k \right )}}{c_1} \\ y &= -\frac {\sqrt {c_1 \left (\left (c_2 +x \right )^{2} c_1^{2}+k \right )}}{c_1} \\ \end{align*}

✓ Mathematica. Time used: 1.512 (sec). Leaf size: 63

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

\begin{align*} y(x)&\to -\frac {\sqrt {k+c_1{}^2 (x+c_2){}^2}}{\sqrt {c_1}}\\ y(x)&\to \frac {\sqrt {k+c_1{}^2 (x+c_2){}^2}}{\sqrt {c_1}}\\ y(x)&\to \text {Indeterminate} \end{align*}

✓ Sympy. Time used: 0.347 (sec). Leaf size: 112

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

\[ \left [ \begin {cases} \frac {i \sqrt {-1 + \frac {k}{C_{1} y^{2}{\left (x \right )}}} y{\left (x \right )}}{\sqrt {C_{1}}} & \text {for}\: \left |{\frac {k}{C_{1} y^{2}{\left (x \right )}}}\right | > 1 \\\frac {\sqrt {1 - \frac {k}{C_{1} y^{2}{\left (x \right )}}} y{\left (x \right )}}{\sqrt {C_{1}}} & \text {otherwise} \end {cases} = C_{1} + x, \ \begin {cases} \frac {i \sqrt {-1 + \frac {k}{C_{1} y^{2}{\left (x \right )}}} y{\left (x \right )}}{\sqrt {C_{1}}} & \text {for}\: \left |{\frac {k}{C_{1} y^{2}{\left (x \right )}}}\right | > 1 \\\frac {\sqrt {1 - \frac {k}{C_{1} y^{2}{\left (x \right )}}} y{\left (x \right )}}{\sqrt {C_{1}}} & \text {otherwise} \end {cases} = C_{1} - x\right ] \]