69.7.1 problem 175

Internal problem ID [18081]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 7, Total differential equations. The integrating factor. Exercises page 61
Problem number : 175
Date solved : Thursday, October 02, 2025 at 02:37:44 PM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

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

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

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

\begin{align*} y &= -\frac {\sqrt {-2 c_1 \,x^{2}-2 \sqrt {-3 x^{4} c_1^{2}+4}}}{2 \sqrt {c_1}} \\ y &= \frac {\sqrt {-2 c_1 \,x^{2}-2 \sqrt {-3 x^{4} c_1^{2}+4}}}{2 \sqrt {c_1}} \\ y &= -\frac {\sqrt {-2 c_1 \,x^{2}+2 \sqrt {-3 x^{4} c_1^{2}+4}}}{2 \sqrt {c_1}} \\ y &= \frac {\sqrt {-2 c_1 \,x^{2}+2 \sqrt {-3 x^{4} c_1^{2}+4}}}{2 \sqrt {c_1}} \\ \end{align*}

✓ Mathematica. Time used: 24.326 (sec). Leaf size: 303

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

\begin{align*} y(x)&\to -\frac {\sqrt {-x^2-\sqrt {-3 x^4+4 e^{2 c_1}}}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {-x^2-\sqrt {-3 x^4+4 e^{2 c_1}}}}{\sqrt {2}}\\ y(x)&\to -\frac {\sqrt {-x^2+\sqrt {-3 x^4+4 e^{2 c_1}}}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {-x^2+\sqrt {-3 x^4+4 e^{2 c_1}}}}{\sqrt {2}}\\ y(x)&\to -\frac {\sqrt {-\sqrt {3} \sqrt {-x^4}-x^2}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {-\sqrt {3} \sqrt {-x^4}-x^2}}{\sqrt {2}}\\ y(x)&\to -\frac {\sqrt {\sqrt {3} \sqrt {-x^4}-x^2}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {\sqrt {3} \sqrt {-x^4}-x^2}}{\sqrt {2}} \end{align*}

✓ Sympy. Time used: 3.121 (sec). Leaf size: 109

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

\[ \left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- x^{2} - \sqrt {C_{1} - 3 x^{4}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- x^{2} - \sqrt {C_{1} - 3 x^{4}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- x^{2} + \sqrt {C_{1} - 3 x^{4}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- x^{2} + \sqrt {C_{1} - 3 x^{4}}}}{2}\right ] \]