89.6.3 problem 3

Internal problem ID [24386]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Miscellaneous Exercises at page 45
Problem number : 3
Date solved : Thursday, October 02, 2025 at 10:22:53 PM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 451

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

\begin{align*} y &= \frac {\frac {\left (108 c_1 x -8 x^{3} c_1^{3}+12 \sqrt {3}\, \sqrt {-x^{2} c_1^{2} \left (4 x^{2} c_1^{2}-27\right )}\right )^{{1}/{3}}}{2}+\frac {2 x^{2} c_1^{2}}{\left (108 c_1 x -8 x^{3} c_1^{3}+12 \sqrt {3}\, \sqrt {-x^{2} c_1^{2} \left (4 x^{2} c_1^{2}-27\right )}\right )^{{1}/{3}}}-c_1 x}{3 c_1} \\ y &= \frac {4 i \sqrt {3}\, c_1^{2} x^{2}-i \sqrt {3}\, \left (108 c_1 x -8 x^{3} c_1^{3}+12 \sqrt {3}\, \sqrt {-4 \left (x^{2} c_1^{2}-\frac {27}{4}\right ) x^{2} c_1^{2}}\right )^{{2}/{3}}-4 x^{2} c_1^{2}-4 c_1 x \left (108 c_1 x -8 x^{3} c_1^{3}+12 \sqrt {3}\, \sqrt {-4 \left (x^{2} c_1^{2}-\frac {27}{4}\right ) x^{2} c_1^{2}}\right )^{{1}/{3}}-\left (108 c_1 x -8 x^{3} c_1^{3}+12 \sqrt {3}\, \sqrt {-4 \left (x^{2} c_1^{2}-\frac {27}{4}\right ) x^{2} c_1^{2}}\right )^{{2}/{3}}}{12 \left (108 c_1 x -8 x^{3} c_1^{3}+12 \sqrt {3}\, \sqrt {-4 \left (x^{2} c_1^{2}-\frac {27}{4}\right ) x^{2} c_1^{2}}\right )^{{1}/{3}} c_1} \\ y &= \frac {i \left (-4 x^{2} c_1^{2}+\left (108 c_1 x -8 x^{3} c_1^{3}+12 \sqrt {3}\, \sqrt {-4 \left (x^{2} c_1^{2}-\frac {27}{4}\right ) x^{2} c_1^{2}}\right )^{{2}/{3}}\right ) \sqrt {3}-{\left (2 c_1 x +\left (108 c_1 x -8 x^{3} c_1^{3}+12 \sqrt {3}\, \sqrt {-4 \left (x^{2} c_1^{2}-\frac {27}{4}\right ) x^{2} c_1^{2}}\right )^{{1}/{3}}\right )}^{2}}{12 \left (108 c_1 x -8 x^{3} c_1^{3}+12 \sqrt {3}\, \sqrt {-4 \left (x^{2} c_1^{2}-\frac {27}{4}\right ) x^{2} c_1^{2}}\right )^{{1}/{3}} c_1} \\ \end{align*}

✓ Mathematica. Time used: 60.101 (sec). Leaf size: 413

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

\begin{align*} y(x)&\to \frac {1}{3} \left (\frac {x^2}{\sqrt [3]{-x^3+\frac {3}{2} \sqrt {3} \sqrt {e^{c_1} x^2 \left (-4 x^2+27 e^{c_1}\right )}+\frac {27 e^{c_1} x}{2}}}+\sqrt [3]{-x^3+\frac {3}{2} \sqrt {3} \sqrt {e^{c_1} x^2 \left (-4 x^2+27 e^{c_1}\right )}+\frac {27 e^{c_1} x}{2}}-x\right )\\ y(x)&\to \frac {1}{12} \left (-\frac {2 \left (1+i \sqrt {3}\right ) x^2}{\sqrt [3]{-x^3+\frac {3}{2} \sqrt {3} \sqrt {e^{c_1} x^2 \left (-4 x^2+27 e^{c_1}\right )}+\frac {27 e^{c_1} x}{2}}}+i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{-2 x^3+3 \sqrt {3} \sqrt {e^{c_1} x^2 \left (-4 x^2+27 e^{c_1}\right )}+27 e^{c_1} x}-4 x\right )\\ y(x)&\to \frac {1}{12} \left (\frac {2 i \left (\sqrt {3}+i\right ) x^2}{\sqrt [3]{-x^3+\frac {3}{2} \sqrt {3} \sqrt {e^{c_1} x^2 \left (-4 x^2+27 e^{c_1}\right )}+\frac {27 e^{c_1} x}{2}}}-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{-2 x^3+3 \sqrt {3} \sqrt {e^{c_1} x^2 \left (-4 x^2+27 e^{c_1}\right )}+27 e^{c_1} x}-4 x\right ) \end{align*}

✓ Sympy. Time used: 28.104 (sec). Leaf size: 428

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

\[ \left [ y{\left (x \right )} = - \frac {\sqrt [3]{2} x^{2}}{3 \sqrt [3]{- 27 C_{1} x + 2 x^{3} + 3 \sqrt {3} \sqrt {C_{1} x^{2} \left (27 C_{1} - 4 x^{2}\right )}}} - \frac {x}{3} - \frac {2^{\frac {2}{3}} \sqrt [3]{- 27 C_{1} x + 2 x^{3} + 3 \sqrt {3} \sqrt {C_{1} x^{2} \left (27 C_{1} - 4 x^{2}\right )}}}{6}, \ y{\left (x \right )} = \frac {\frac {4 \sqrt [3]{2} x^{2}}{\sqrt [3]{- 27 C_{1} x + 2 x^{3} + 3 \sqrt {3} \sqrt {C_{1} x^{2} \left (27 C_{1} - 4 x^{2}\right )}}} - 2 x + 2 \sqrt {3} i x - 2^{\frac {2}{3}} \sqrt [3]{- 27 C_{1} x + 2 x^{3} + 3 \sqrt {3} \sqrt {C_{1} x^{2} \left (27 C_{1} - 4 x^{2}\right )}} - 2^{\frac {2}{3}} \sqrt {3} i \sqrt [3]{- 27 C_{1} x + 2 x^{3} + 3 \sqrt {3} \sqrt {C_{1} x^{2} \left (27 C_{1} - 4 x^{2}\right )}}}{6 \left (1 - \sqrt {3} i\right )}, \ y{\left (x \right )} = \frac {\frac {4 \sqrt [3]{2} x^{2}}{\sqrt [3]{- 27 C_{1} x + 2 x^{3} + 3 \sqrt {3} \sqrt {C_{1} x^{2} \left (27 C_{1} - 4 x^{2}\right )}}} - 2 x - 2 \sqrt {3} i x - 2^{\frac {2}{3}} \sqrt [3]{- 27 C_{1} x + 2 x^{3} + 3 \sqrt {3} \sqrt {C_{1} x^{2} \left (27 C_{1} - 4 x^{2}\right )}} + 2^{\frac {2}{3}} \sqrt {3} i \sqrt [3]{- 27 C_{1} x + 2 x^{3} + 3 \sqrt {3} \sqrt {C_{1} x^{2} \left (27 C_{1} - 4 x^{2}\right )}}}{6 \left (1 + \sqrt {3} i\right )}\right ] \]