25.3.5 problem 5.3

Internal problem ID [6870]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 4
Problem number : 5.3
Date solved : Tuesday, September 30, 2025 at 03:58:56 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.124 (sec). Leaf size: 55

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

\begin{align*} y &= \frac {1-\sqrt {-4 c_1^{2} x^{2}+4 c_1 x +1}}{2 c_1} \\ y &= \frac {1+\sqrt {-4 c_1^{2} x^{2}+4 c_1 x +1}}{2 c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.777 (sec). Leaf size: 75

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

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

✓ Sympy. Time used: 1.473 (sec). Leaf size: 58

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

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