83.9.2 problem 2

Internal problem ID [21960]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter III. First order differential equations of the first degree. Ex. X at page 57
Problem number : 2
Date solved : Thursday, October 02, 2025 at 08:20:03 PM
CAS classification : [_separable]

\begin{align*} 4 x^{2} y^{2} y^{\prime }-3 x y^{3}&=x^{2} y^{3}+2 x^{2} y^{\prime } \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 51

ode:=4*x^2*y(x)^2*diff(y(x),x)-3*x*y(x)^3 = x^2*y(x)^3+2*x^2*diff(y(x),x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {x^{{3}/{4}} {\mathrm e}^{\frac {c_1}{4}+\frac {x}{4}} \sqrt {2}\, \sqrt {-\frac {{\mathrm e}^{-\frac {c_1}{2}-\frac {x}{2}}}{x^{{3}/{2}} \operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {c_1}{2}-\frac {x}{2}}}{2 x^{{3}/{2}}}\right )}}}{2} \]

✓ Mathematica. Time used: 3.711 (sec). Leaf size: 84

ode=4*x^2*y[x]^2*D[y[x],x]-3*x*y[x]^3== x^2*y[x]^3+2*x^2*D[y[x],x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {i}{\sqrt {2} \sqrt {W\left (-\frac {e^{-\frac {x}{2}-c_1}}{2 x^{3/2}}\right )}}\\ y(x)&\to \frac {i}{\sqrt {2} \sqrt {W\left (-\frac {e^{-\frac {x}{2}-c_1}}{2 x^{3/2}}\right )}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 1.109 (sec). Leaf size: 70

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2*y(x)**3 + 4*x**2*y(x)**2*Derivative(y(x), x) - 2*x**2*Derivative(y(x), x) - 3*x*y(x)**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = x^{\frac {3}{4}} e^{\frac {C_{1}}{2} + \frac {x}{4} + \frac {W\left (- \frac {e^{- C_{1} - \frac {x}{2}}}{2 x^{\frac {3}{2}}}\right )}{2}}, \ y{\left (x \right )} = x^{\frac {3}{4}} e^{\frac {C_{1}}{2} + \frac {x}{4} + \frac {W\left (\frac {e^{- C_{1} - \frac {x}{2}}}{2 x^{\frac {3}{2}}}\right )}{2}}\right ] \]