23.1.644 problem 638

Internal problem ID [5251]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 638
Date solved : Tuesday, September 30, 2025 at 12:01:47 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.075 (sec). Leaf size: 59

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

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

✓ Mathematica. Time used: 2.574 (sec). Leaf size: 87

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

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

✓ Sympy. Time used: 2.058 (sec). Leaf size: 70

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

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