56.8.3 problem Ex 3

Internal problem ID [14112]
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 15. Page 22
Problem number : Ex 3
Date solved : Thursday, October 02, 2025 at 09:14:34 AM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} 2 x^{3} y-y^{2}-\left (2 x^{4}+y x \right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.157 (sec). Leaf size: 47

ode:=2*x^3*y(x)-y(x)^2-(2*x^4+x*y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\left (c_1 +\sqrt {4 x^{4}+c_1^{2}}\right ) c_1}{2 x} \\ y &= -\frac {\left (-c_1 +\sqrt {4 x^{4}+c_1^{2}}\right ) c_1}{2 x} \\ \end{align*}

✓ Mathematica. Time used: 0.605 (sec). Leaf size: 76

ode=(2*x^3*y[x]-y[x]^2)-(2*x^4+x*y[x])*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {2 x^4}{-x+\frac {\sqrt {1+4 c_1 x^4}}{\sqrt {\frac {1}{x^2}}}}\\ y(x)&\to -\frac {2 x^4}{x+\frac {\sqrt {1+4 c_1 x^4}}{\sqrt {\frac {1}{x^2}}}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 5.677 (sec). Leaf size: 48

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

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