54.7.37 problem 1632 (6.42)

Internal problem ID [12886]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1632 (6.42)
Date solved : Wednesday, October 01, 2025 at 02:44:42 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 }-2 a y y^{\prime }&=0 \end{align*}

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

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

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

✓ Mathematica. Time used: 0.105 (sec). Leaf size: 90

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-2*a*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))/(2*a*y(x)) cannot be solved by the factorable group method