89.2.6 problem 6

Internal problem ID [24271]
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. Exercises at page 27
Problem number : 6
Date solved : Thursday, October 02, 2025 at 10:04:07 PM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} \left (x -y\right ) \left (4 x +y\right )+x \left (5 x -y\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.017 (sec). Leaf size: 69

ode:=(x-y(x))*(4*x+y(x))+x*(5*x-y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {-2 x^{2} c_1^{2}+\sqrt {-12 x^{2} c_1^{2}+1}+1}{2 c_1^{2} x} \\ y &= \frac {-2 x^{2} c_1^{2}-\sqrt {-12 x^{2} c_1^{2}+1}+1}{2 c_1^{2} x} \\ \end{align*}

✓ Mathematica. Time used: 0.713 (sec). Leaf size: 90

ode=(x-y[x])*(4*x+y[x])+x*(5*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^2-e^{\frac {c_1}{2}} \sqrt {-12 x^2+e^{c_1}}+e^{c_1}}{2 x}\\ y(x)&\to \frac {-2 x^2+e^{\frac {c_1}{2}} \sqrt {-12 x^2+e^{c_1}}+e^{c_1}}{2 x} \end{align*}

✓ Sympy. Time used: 1.700 (sec). Leaf size: 48

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(5*x - y(x))*Derivative(y(x), x) + (x - y(x))*(4*x + y(x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {C_{1}}{2 x} - x - \frac {\sqrt {C_{1} \left (C_{1} - 12 x^{2}\right )}}{2 x}, \ y{\left (x \right )} = \frac {C_{1}}{2 x} - x + \frac {\sqrt {C_{1} \left (C_{1} - 12 x^{2}\right )}}{2 x}\right ] \]