69.7.21 problem 196

Internal problem ID [18101]
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 : 196
Date solved : Thursday, October 02, 2025 at 02:42:13 PM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} 3 y^{2}-x +\left (2 y^{3}-6 y x \right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.037 (sec). Leaf size: 101

ode:=3*y(x)^2-x+(2*y(x)^3-6*x*y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {\sqrt {-2 \sqrt {c_1 \left (c_1 -8 x \right )}+2 c_1 -4 x}}{2} \\ y &= \frac {\sqrt {-2 \sqrt {c_1 \left (c_1 -8 x \right )}+2 c_1 -4 x}}{2} \\ y &= -\frac {\sqrt {2 \sqrt {c_1 \left (c_1 -8 x \right )}+2 c_1 -4 x}}{2} \\ y &= \frac {\sqrt {2 \sqrt {c_1 \left (c_1 -8 x \right )}+2 c_1 -4 x}}{2} \\ \end{align*}

✓ Mathematica. Time used: 8.997 (sec). Leaf size: 185

ode=( 3*y[x]^2-x)+( 2*y[x]^3-6*x*y[x] )*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 8.943 (sec). Leaf size: 160

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x + (-6*x*y(x) + 2*y(x)**3)*Derivative(y(x), x) + 3*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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