89.6.35 problem 36

Internal problem ID [24418]
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 : 36
Date solved : Thursday, October 02, 2025 at 10:26:35 PM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 41

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

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

✓ Mathematica. Time used: 0.644 (sec). Leaf size: 65

ode=y[x]^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 {x-\sqrt {x^2+4 c_1}}{2 c_1}\\ y(x)&\to \frac {x+\sqrt {x^2+4 c_1}}{2 c_1}\\ y(x)&\to 0\\ y(x)&\to \text {Indeterminate} \end{align*}

✓ Sympy. Time used: 0.815 (sec). Leaf size: 41

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

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