54.7.1 problem 1591 (6.1)

Internal problem ID [12850]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1591 (6.1)
Date solved : Wednesday, October 01, 2025 at 02:22:31 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} y^{\prime \prime }-y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.015 (sec). Leaf size: 12

ode:=diff(diff(y(x),x),x)-y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = 6 \operatorname {WeierstrassP}\left (x +c_1 , 0, c_2\right ) \]

✓ Mathematica. Time used: 0.512 (sec). Leaf size: 26

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

\begin{align*} y(x)&\to \sqrt [3]{6} \wp \left (\frac {x+c_1}{\sqrt [3]{6}};0,c_2\right ) \end{align*}

✓ Sympy. Time used: 6.584 (sec). Leaf size: 88

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

\[ \left [ \frac {y{\left (x \right )} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {2 e^{i \pi } y^{3}{\left (x \right )}}{3 C_{1}}} \right )}}{3 \sqrt {C_{1}} \Gamma \left (\frac {4}{3}\right )} = C_{2} + x, \ \frac {y{\left (x \right )} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {2 e^{i \pi } y^{3}{\left (x \right )}}{3 C_{1}}} \right )}}{3 \sqrt {C_{1}} \Gamma \left (\frac {4}{3}\right )} = C_{2} - x\right ] \]