54.7.180 problem 1800 (book 6.209)

Internal problem ID [13029]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1800 (book 6.209)
Date solved : Wednesday, October 01, 2025 at 02:57:21 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} y^{3} y^{\prime \prime }-a&=0 \end{align*}

✓ Maple. Time used: 0.151 (sec). Leaf size: 46

ode:=y(x)^3*diff(diff(y(x),x),x)-a = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 1.592 (sec). Leaf size: 63

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

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

✓ Sympy. Time used: 0.348 (sec). Leaf size: 112

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

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