89.7.15 problem 15

Internal problem ID [24446]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 4. Additional topics on equations of first order and first degree. Exercises at page 61
Problem number : 15
Date solved : Thursday, October 02, 2025 at 10:35:16 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} 2 y+x \left (x^{2} \ln \left (y\right )-1\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.066 (sec). Leaf size: 34

ode:=2*y(x)+x*(x^2*ln(y(x))-1)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{\frac {\operatorname {LambertW}\left (-c_1 \,{\mathrm e}^{\frac {-x^{2}-1}{x^{2}}}\right ) x^{2}+x^{2}+1}{x^{2}}} \]

✓ Mathematica. Time used: 60.121 (sec). Leaf size: 23

ode=(2*y[x])+x*(x^2*Log[y[x]]-1)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {c_1}{W\left (c_1 e^{-\frac {1}{x^2}-1}\right )} \end{align*}

✓ Sympy. Time used: 0.707 (sec). Leaf size: 27

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

\[ y{\left (x \right )} = e^{W\left (- 2 C_{1} e^{-1 - \frac {1}{x^{2}}}\right ) + 1 + \frac {1}{x^{2}}} \]