81.1.40 problem 2-38

Internal problem ID [21485]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 2. Separable differential equations
Problem number : 2-38
Date solved : Thursday, October 02, 2025 at 07:41:38 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.158 (sec). Leaf size: 115

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

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

✓ Mathematica. Time used: 6.91 (sec). Leaf size: 218

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

\begin{align*} y(x)&\to -\sqrt {-x^2-\sqrt {x^4+e^{4 c_1}}}\\ y(x)&\to \sqrt {-x^2-\sqrt {x^4+e^{4 c_1}}}\\ y(x)&\to -\sqrt {-x^2+\sqrt {x^4+e^{4 c_1}}}\\ y(x)&\to \sqrt {-x^2+\sqrt {x^4+e^{4 c_1}}}\\ y(x)&\to 0\\ y(x)&\to -\sqrt {-\sqrt {x^4}-x^2}\\ y(x)&\to \sqrt {-\sqrt {x^4}-x^2}\\ y(x)&\to -\sqrt {\sqrt {x^4}-x^2}\\ y(x)&\to \sqrt {\sqrt {x^4}-x^2} \end{align*}

✓ Sympy. Time used: 2.397 (sec). Leaf size: 75

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

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