56.3.2 problem Ex 2

Internal problem ID [14089]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 10. Homogeneous equations. Page 15
Problem number : Ex 2
Date solved : Thursday, October 02, 2025 at 09:11:33 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.142 (sec). Leaf size: 89

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

\begin{align*} y &= -\frac {\sqrt {-2-2 \sqrt {4 c_1 \,x^{2}+1}}\, x}{2} \\ y &= \frac {\sqrt {-2-2 \sqrt {4 c_1 \,x^{2}+1}}\, x}{2} \\ y &= -\frac {\sqrt {-2+2 \sqrt {4 c_1 \,x^{2}+1}}\, x}{2} \\ y &= \frac {\sqrt {-2+2 \sqrt {4 c_1 \,x^{2}+1}}\, x}{2} \\ \end{align*}

✓ Mathematica. Time used: 39.613 (sec). Leaf size: 277

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

\begin{align*} y(x)&\to -\frac {\sqrt {-x^2-\sqrt {x^4+4 e^{2 c_1} x^6}}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {-x^2-\sqrt {x^4+4 e^{2 c_1} x^6}}}{\sqrt {2}}\\ y(x)&\to -\frac {\sqrt {-x^2+\sqrt {x^4+4 e^{2 c_1} x^6}}}{\sqrt {2}}\\ y(x)&\to \sqrt {-\frac {x^2}{2}+\frac {1}{2} \sqrt {x^4+4 e^{2 c_1} x^6}}\\ y(x)&\to -\frac {\sqrt {-\sqrt {x^4}-x^2}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {-\sqrt {x^4}-x^2}}{\sqrt {2}}\\ y(x)&\to -\frac {\sqrt {\sqrt {x^4}-x^2}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {\sqrt {x^4}-x^2}}{\sqrt {2}} \end{align*}

✓ Sympy. Time used: 4.798 (sec). Leaf size: 116

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

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