43.23.5 problem 1(e)

Internal problem ID [9049]
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(e)
Date solved : Tuesday, September 30, 2025 at 06:02:27 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.050 (sec). Leaf size: 23

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

\[ y = \frac {\tan \left (\frac {\left (x +c_2 \right ) \sqrt {2}}{2 c_1}\right ) \sqrt {2}}{c_1} \]

✓ Mathematica. Time used: 0.07 (sec). Leaf size: 96

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\frac {K[1]^2}{2}+c_1}dK[1]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\frac {K[1]^2}{2}-c_1}dK[1]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\frac {K[1]^2}{2}+c_1}dK[1]\&\right ][x+c_2] \end{align*}

✗ Sympy

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

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