56.1.2 problem Ex 2

Internal problem ID [14080]
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 8. Exact differential equations. Page 11
Problem number : Ex 2
Date solved : Thursday, October 02, 2025 at 09:10:44 AM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

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

✓ Maple. Time used: 0.113 (sec). Leaf size: 71

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

\begin{align*} y &= \frac {\sqrt {\frac {2 c_1 \,x^{3}-2 \sqrt {c_1^{2} x^{6}+4}}{c_1 \,x^{3}}}\, x}{2} \\ y &= \frac {\sqrt {2}\, \sqrt {\frac {c_1 \,x^{3}+\sqrt {c_1^{2} x^{6}+4}}{c_1 \,x^{3}}}\, x}{2} \\ \end{align*}

✓ Mathematica. Time used: 9.65 (sec). Leaf size: 277

ode=(y[x]^2-2*x^2)/(x*y[x]^2-x^3)+ (2*y[x]^2-x^2)/(y[x]^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 {\sqrt {x^2-\frac {\sqrt {x^6-4 e^{2 c_1}}}{x}}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {x^2-\frac {\sqrt {x^6-4 e^{2 c_1}}}{x}}}{\sqrt {2}}\\ y(x)&\to -\frac {\sqrt {\frac {x^3+\sqrt {x^6-4 e^{2 c_1}}}{x}}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {\frac {x^3+\sqrt {x^6-4 e^{2 c_1}}}{x}}}{\sqrt {2}}\\ y(x)&\to -\frac {\sqrt {x^2-\frac {\sqrt {x^6}}{x}}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {x^2-\frac {\sqrt {x^6}}{x}}}{\sqrt {2}}\\ y(x)&\to -\frac {\sqrt {\frac {\sqrt {x^6}+x^3}{x}}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {\frac {\sqrt {x^6}+x^3}{x}}}{\sqrt {2}} \end{align*}

✓ Sympy. Time used: 3.772 (sec). Leaf size: 105

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

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