60.7.61 problem 1669 (book 6.78)

Internal problem ID [11611]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1669 (book 6.78)
Date solved : Sunday, March 30, 2025 at 08:31:31 PM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x y^{\prime \prime }+\left (y-1\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.061 (sec). Leaf size: 24

ode:=x*diff(diff(y(x),x),x)+(-1+y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.149 (sec). Leaf size: 120

ode=(-1 + y[x])*D[y[x],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 {$\#$1}}\frac {1}{K[1]^2-4 K[1]-2 c_1+4}dK[1]\&\right ]\left [-\frac {\log (x)}{2}+c_2\right ] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1]^2-4 K[1]-2 (-1) c_1+4}dK[1]\&\right ]\left [-\frac {\log (x)}{2}+c_2\right ] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1]^2-4 K[1]-2 c_1+4}dK[1]\&\right ]\left [-\frac {\log (x)}{2}+c_2\right ] \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + (y(x) - 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE x*Derivative(y(x), (x, 2))/(y(x) - 1) + Derivative(y(x), x) cannot be solved by the factorable group method