62.37.3 problem Ex 3

Internal problem ID [12939]
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 : Monday, March 31, 2025 at 07:27:32 AM
CAS classification : [[_2nd_order, _reducible, _mu_xy]]

\begin{align*} y y^{\prime \prime }-{y^{\prime }}^{2}&=y^{2} \ln \left (y\right )-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: 60.199 (sec). Leaf size: 275

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]
 

\[ y(x)\to -\frac {\exp \left (-\int _1^x\frac {e^{-K[3]} \left (y(K[3]) \left (e^{2 K[3]} c_1{}^2-e^{K[3]} c_1-K[3]^2+e^{2 K[3]} \int _1^{K[3]}\frac {e^{-K[1]} \left (\left (K[1]^2-\log (y(K[1]))\right ) y(K[1])+y''(K[1])\right )}{y(K[1])}dK[1]{}^2+\log (y(K[3]))+e^{K[3]} \left (2 e^{K[3]} c_1-1\right ) \int _1^{K[3]}\frac {e^{-K[1]} \left (\left (K[1]^2-\log (y(K[1]))\right ) y(K[1])+y''(K[1])\right )}{y(K[1])}dK[1]\right )-y''(K[3])\right )}{y(K[3]) \left (c_1+\int _1^{K[3]}\frac {e^{-K[1]} \left (\left (K[1]^2-\log (y(K[1]))\right ) y(K[1])+y''(K[1])\right )}{y(K[1])}dK[1]\right )}dK[3]-x+c_2\right )}{\int _1^x\frac {e^{-K[1]} \left (\left (K[1]^2-\log (y(K[1]))\right ) y(K[1])+y''(K[1])\right )}{y(K[1])}dK[1]+c_1} \]

✗ 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