56.37.3 problem Ex 3

Internal problem ID [14291]
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 3
Date solved : Friday, October 03, 2025 at 07:29:39 AM
CAS classification : [[_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 22

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

\[ y = {\mathrm e}^{x^{2}+2-\frac {{\mathrm e}^{x} c_1}{2}+\frac {{\mathrm e}^{-x} c_2}{2}} \]

✓ Mathematica. Time used: 0.383 (sec). Leaf size: 30

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

\begin{align*} y(x)&\to e^{x^2-\frac {c_1 e^x}{2}-c_2 e^{-x}+2} \end{align*}

✗ Sympy

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

TypeError : cannot determine truth value of Relational: _X0**2 < 2