43.23.3 problem 1(c)

Internal problem ID [9047]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 6. Existence and uniqueness of solutions to systems and nth order equations. Page 238
Problem number : 1(c)
Date solved : Tuesday, September 30, 2025 at 06:02:25 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y y^{\prime \prime }+4 {y^{\prime }}^{2}&=0 \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 154

ode:=y(x)*diff(diff(y(x),x),x)+4*diff(y(x),x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= \left (5 c_1 x +5 c_2 \right )^{{1}/{5}} \\ y &= -\frac {\left (i \sqrt {2}\, \sqrt {5-\sqrt {5}}+\sqrt {5}+1\right ) \left (5 c_1 x +5 c_2 \right )^{{1}/{5}}}{4} \\ y &= \frac {\left (i \sqrt {2}\, \sqrt {5-\sqrt {5}}-\sqrt {5}-1\right ) \left (5 c_1 x +5 c_2 \right )^{{1}/{5}}}{4} \\ y &= -\frac {\left (i \sqrt {2}\, \sqrt {5+\sqrt {5}}-\sqrt {5}+1\right ) \left (5 c_1 x +5 c_2 \right )^{{1}/{5}}}{4} \\ y &= \frac {\left (i \sqrt {2}\, \sqrt {5+\sqrt {5}}+\sqrt {5}-1\right ) \left (5 c_1 x +5 c_2 \right )^{{1}/{5}}}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.108 (sec). Leaf size: 20

ode=y[x]*D[y[x],{x,2}]+4*(D[y[x],x])^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2 \sqrt [5]{5 x-c_1} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)*Derivative(y(x), (x, 2)) + 4*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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