87.7.10 problem 13

Internal problem ID [23351]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 1. Elementary methods. First order differential equations. Exercise at page 57
Problem number : 13
Date solved : Thursday, October 02, 2025 at 09:39:27 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y^{\prime \prime }-\frac {2 y^{\prime }}{y^{3}}&=0 \end{align*}

✓ Maple. Time used: 0.025 (sec). Leaf size: 28

ode:=diff(diff(y(x),x),x)-2/y(x)^3*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {y}{c_1^{2}}-\frac {\operatorname {arctanh}\left (c_1 y\right )}{\left (c_1^{2}\right )^{{3}/{2}}}-x -c_2 = 0 \]

✓ Mathematica. Time used: 0.854 (sec). Leaf size: 115

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\frac {\text {$\#$1}}{c_1}-\frac {\text {arctanh}\left (\text {$\#$1} \sqrt {c_1}\right )}{c_1{}^{3/2}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [-\frac {\text {arctanh}\left (\text {$\#$1} \sqrt {-c_1}\right )}{(-c_1){}^{3/2}}-\frac {\text {$\#$1}}{c_1}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\frac {\text {$\#$1}}{c_1}-\frac {\text {arctanh}\left (\text {$\#$1} \sqrt {c_1}\right )}{c_1{}^{3/2}}\&\right ][x+c_2] \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -y(x)**3*Derivative(y(x), (x, 2))/2 + Derivative(y(x), x) cannot