75.7.13 problem 188

Internal problem ID [16735]
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 : 188
Date solved : Monday, March 31, 2025 at 03:15:03 PM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 270

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

\begin{align*} y &= -\frac {12^{{1}/{3}} \left (c_1^{2} x^{4} 12^{{1}/{3}}-{\left (\left (\sqrt {3}\, \sqrt {4 c_1^{4} x^{8}+27}+9\right ) x^{2} c_1 \right )}^{{2}/{3}}\right )}{6 c_1 x {\left (\left (\sqrt {3}\, \sqrt {4 c_1^{4} x^{8}+27}+9\right ) x^{2} c_1 \right )}^{{1}/{3}}} \\ y &= -\frac {\left (\left (1+i \sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {4 c_1^{4} x^{8}+27}+9\right ) x^{2} c_1 \right )}^{{2}/{3}}+\left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) c_1^{2} x^{4} 2^{{2}/{3}}\right ) 3^{{1}/{3}} 2^{{2}/{3}}}{12 {\left (\left (\sqrt {3}\, \sqrt {4 c_1^{4} x^{8}+27}+9\right ) x^{2} c_1 \right )}^{{1}/{3}} c_1 x} \\ y &= \frac {\left (\left (i \sqrt {3}-1\right ) {\left (\left (\sqrt {3}\, \sqrt {4 c_1^{4} x^{8}+27}+9\right ) x^{2} c_1 \right )}^{{2}/{3}}+c_1^{2} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) x^{4} 2^{{2}/{3}}\right ) 3^{{1}/{3}} 2^{{2}/{3}}}{12 {\left (\left (\sqrt {3}\, \sqrt {4 c_1^{4} x^{8}+27}+9\right ) x^{2} c_1 \right )}^{{1}/{3}} c_1 x} \\ \end{align*}

✓ Mathematica. Time used: 0.118 (sec). Leaf size: 44

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

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {3 K[1]^2+1}{K[1] \left (K[1]^2+1\right )}dK[1]=-4 \log (x)+c_1,y(x)\right ] \]

✗ Sympy

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

Timed Out