83.7.1 problem 1

Internal problem ID [19048]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter II. Equations of first order and first degree. Exercise II (F) at page 24
Problem number : 1
Date solved : Monday, March 31, 2025 at 06:36:47 PM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 19

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

\[ y = -\frac {x}{3 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{\frac {c_1}{3}} x^{{1}/{3}}}{3}\right )} \]

✓ Mathematica. Time used: 4.432 (sec). Leaf size: 35

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

\begin{align*} y(x)\to -\frac {x}{3 W\left (-\frac {1}{3} e^{-\frac {c_1}{3}} \sqrt [3]{x}\right )} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 14.842 (sec). Leaf size: 90

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

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