89.2.8 problem 8

Internal problem ID [24273]
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 : 8
Date solved : Thursday, October 02, 2025 at 10:05:25 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} x^{2}+2 y x -4 y^{2}-\left (x^{2}-8 y x -4 y^{2}\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.060 (sec). Leaf size: 55

ode:=x^2+2*x*y(x)-4*y(x)^2-(x^2-8*x*y(x)-4*y(x)^2)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {1-\sqrt {-16 c_1^{2} x^{2}+16 c_1 x +1}}{8 c_1} \\ y &= \frac {1+\sqrt {-16 c_1^{2} x^{2}+16 c_1 x +1}}{8 c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.838 (sec). Leaf size: 75

ode=(x^2+2*x*y[x]-4*y[x]^2)-(x^2-8*x*y[x]-4*y[x]^2)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{8} \left (e^{c_1}-\sqrt {-16 x^2+16 e^{c_1} x+e^{2 c_1}}\right )\\ y(x)&\to \frac {1}{8} \left (\sqrt {-16 x^2+16 e^{c_1} x+e^{2 c_1}}+e^{c_1}\right ) \end{align*}

✓ Sympy. Time used: 1.452 (sec). Leaf size: 58

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2 + 2*x*y(x) - (x**2 - 8*x*y(x) - 4*y(x)**2)*Derivative(y(x), x) - 4*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \frac {\sqrt {- 16 x^{2} + 16 x e^{C_{1}} + e^{2 C_{1}}}}{8} + \frac {e^{C_{1}}}{8}, \ y{\left (x \right )} = \frac {\sqrt {- 16 x^{2} + 16 x e^{C_{1}} + e^{2 C_{1}}}}{8} + \frac {e^{C_{1}}}{8}\right ] \]