75.12.4 problem 278

Internal problem ID [16791]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 12. Miscellaneous problems. Exercises page 93
Problem number : 278
Date solved : Thursday, March 13, 2025 at 08:47:16 AM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

\begin{align*} x^{3}-3 x y^{2}+\left (y^{3}-3 x^{2} y\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.361 (sec). Leaf size: 119

ode:=x^3-3*x*y(x)^2+(y(x)^3-3*x^2*y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\sqrt {3 c_{1} x^{2}-\sqrt {8 c_{1}^{2} x^{4}+1}}}{\sqrt {c_{1}}} \\ y &= \frac {\sqrt {3 c_{1} x^{2}+\sqrt {8 c_{1}^{2} x^{4}+1}}}{\sqrt {c_{1}}} \\ y &= -\frac {\sqrt {3 c_{1} x^{2}-\sqrt {8 c_{1}^{2} x^{4}+1}}}{\sqrt {c_{1}}} \\ y &= -\frac {\sqrt {3 c_{1} x^{2}+\sqrt {8 c_{1}^{2} x^{4}+1}}}{\sqrt {c_{1}}} \\ \end{align*}

✓ Mathematica. Time used: 7.925 (sec). Leaf size: 245

ode=(x^3-3*x*y[x]^2)+(y[x]^3-3*x^2*y[x])*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\sqrt {3 x^2-\sqrt {8 x^4+e^{4 c_1}}} \\ y(x)\to \sqrt {3 x^2-\sqrt {8 x^4+e^{4 c_1}}} \\ y(x)\to -\sqrt {3 x^2+\sqrt {8 x^4+e^{4 c_1}}} \\ y(x)\to \sqrt {3 x^2+\sqrt {8 x^4+e^{4 c_1}}} \\ y(x)\to -\sqrt {3 x^2-2 \sqrt {2} \sqrt {x^4}} \\ y(x)\to \sqrt {3 x^2-2 \sqrt {2} \sqrt {x^4}} \\ y(x)\to -\sqrt {2 \sqrt {2} \sqrt {x^4}+3 x^2} \\ y(x)\to \sqrt {2 \sqrt {2} \sqrt {x^4}+3 x^2} \\ \end{align*}

✓ Sympy. Time used: 3.815 (sec). Leaf size: 85

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3 - 3*x*y(x)**2 + (-3*x**2*y(x) + y(x)**3)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \sqrt {3 x^{2} - \sqrt {C_{1} + 8 x^{4}}}, \ y{\left (x \right )} = \sqrt {3 x^{2} - \sqrt {C_{1} + 8 x^{4}}}, \ y{\left (x \right )} = - \sqrt {3 x^{2} + \sqrt {C_{1} + 8 x^{4}}}, \ y{\left (x \right )} = \sqrt {3 x^{2} + \sqrt {C_{1} + 8 x^{4}}}\right ] \]