62.38.3 problem Ex 3

Internal problem ID [12943]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 62. Summary. Page 144
Problem number : Ex 3
Date solved : Monday, March 31, 2025 at 07:27:40 AM
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 }&=0 \end{align*}

✓ Maple. Time used: 0.051 (sec). Leaf size: 23

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

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

✓ Mathematica. Time used: 0.137 (sec). Leaf size: 96

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

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