23.1.694 problem 689

Internal problem ID [5301]
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 : 689
Date solved : Tuesday, September 30, 2025 at 12:19:31 PM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

\begin{align*} \left (5 x^{2}+2 y^{2}\right ) y y^{\prime }+x \left (x^{2}+5 y^{2}\right )&=0 \end{align*}

✓ Maple. Time used: 0.026 (sec). Leaf size: 125

ode:=(5*x^2+2*y(x)^2)*y(x)*diff(y(x),x)+x*(x^2+5*y(x)^2) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {\sqrt {-10 c_1 \,x^{2}-2 \sqrt {23 x^{4} c_1^{2}+2}}}{2 \sqrt {c_1}} \\ y &= \frac {\sqrt {-10 c_1 \,x^{2}-2 \sqrt {23 x^{4} c_1^{2}+2}}}{2 \sqrt {c_1}} \\ y &= -\frac {\sqrt {-10 c_1 \,x^{2}+2 \sqrt {23 x^{4} c_1^{2}+2}}}{2 \sqrt {c_1}} \\ y &= \frac {\sqrt {-10 c_1 \,x^{2}+2 \sqrt {23 x^{4} c_1^{2}+2}}}{2 \sqrt {c_1}} \\ \end{align*}

✓ Mathematica. Time used: 27.32 (sec). Leaf size: 295

ode=(5*x^2+2*y[x]^2)*y[x]*D[y[x],x]+x*(x^2+5*y[x]^2)==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {\sqrt {-5 x^2-\sqrt {23 x^4+2 e^{4 c_1}}}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {-5 x^2-\sqrt {23 x^4+2 e^{4 c_1}}}}{\sqrt {2}}\\ y(x)&\to -\frac {\sqrt {-5 x^2+\sqrt {23 x^4+2 e^{4 c_1}}}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {-5 x^2+\sqrt {23 x^4+2 e^{4 c_1}}}}{\sqrt {2}}\\ y(x)&\to -\frac {\sqrt {-\sqrt {23} \sqrt {x^4}-5 x^2}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {-\sqrt {23} \sqrt {x^4}-5 x^2}}{\sqrt {2}}\\ y(x)&\to -\frac {\sqrt {\sqrt {23} \sqrt {x^4}-5 x^2}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {\sqrt {23} \sqrt {x^4}-5 x^2}}{\sqrt {2}} \end{align*}

✓ Sympy. Time used: 3.060 (sec). Leaf size: 116

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x**2 + 5*y(x)**2) + (5*x**2 + 2*y(x)**2)*y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- 5 x^{2} - \sqrt {C_{1} + 23 x^{4}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- 5 x^{2} - \sqrt {C_{1} + 23 x^{4}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- 5 x^{2} + \sqrt {C_{1} + 23 x^{4}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- 5 x^{2} + \sqrt {C_{1} + 23 x^{4}}}}{2}\right ] \]