78.4.18 problem 19

Internal problem ID [18062]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 2. First order equations. Section 8 (Exact Equations). Problems at page 72
Problem number : 19
Date solved : Thursday, March 13, 2025 at 11:28:01 AM
CAS classification : [_exact, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

\begin{align*} 3 x^{2} \left (1+\ln \left (y\right )\right )+\left (\frac {x^{3}}{y}-2 y\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 43

ode:=3*x^2*(1+ln(y(x)))+(x^3/y(x)-2*y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {x^{3} \operatorname {LambertW}\left (-\frac {2 \,{\mathrm e}^{\frac {-2 x^{3}-2 c_1}{x^{3}}}}{x^{3}}\right )+2 x^{3}+2 c_1}{2 x^{3}}} \]

✓ Mathematica. Time used: 60.172 (sec). Leaf size: 79

ode=(3*x^2*(1+Log[y[x]])) +( x^3/y[x]-2*y[x] )*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {i x^{3/2} \sqrt {W\left (-\frac {2 e^{-2+\frac {2 c_1}{x^3}}}{x^3}\right )}}{\sqrt {2}} \\ y(x)\to \frac {i x^{3/2} \sqrt {W\left (-\frac {2 e^{-2+\frac {2 c_1}{x^3}}}{x^3}\right )}}{\sqrt {2}} \\ \end{align*}

✓ Sympy. Time used: 3.148 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x**2*(log(y(x)) + 1) + (x**3/y(x) - 2*y(x))*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = e^{- \frac {3 C_{1}}{x^{3}} - \frac {W\left (- \frac {2 e^{- \frac {6 C_{1}}{x^{3}} - 2}}{x^{3}}\right )}{2} - 1} \]