62.37.2 problem Ex 2

Internal problem ID [12938]
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 61. Transformation of variables. Page 143
Problem number : Ex 2
Date solved : Monday, March 31, 2025 at 07:27:31 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x^{3} y^{\prime \prime }-\left (x y^{\prime }-y\right )^{2}&=0 \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 19

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

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

✓ Mathematica. Time used: 0.774 (sec). Leaf size: 21

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

\[ y(x)\to -x \log \left (-\frac {c_2 x+c_1}{x}\right ) \]

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (x*y(x) + sqrt(x**5*Derivative(y(x), (x, 2))))/x**2 cannot be solved by the factorable group method