75.5.20 problem 119

Internal problem ID [16685]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 5. Homogeneous equations. Exercises page 44
Problem number : 119
Date solved : Monday, March 31, 2025 at 03:05:29 PM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Maple. Time used: 0.116 (sec). Leaf size: 35

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

\[ \ln \left (x \right )-c_1 +\frac {\ln \left (\frac {y^{6}+x^{2}}{x^{2}}\right )}{2}-\arctan \left (\frac {y^{3}}{x}\right ) = 0 \]

✓ Mathematica. Time used: 0.173 (sec). Leaf size: 117

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

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {3 \left (K[2]^5-x K[2]^2\right )}{K[2]^6+x^2}-\int _1^x\left (\frac {3 K[2]^2}{K[2]^6+K[1]^2}-\frac {6 K[2]^5 \left (K[2]^3+K[1]\right )}{\left (K[2]^6+K[1]^2\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\frac {y(x)^3+K[1]}{y(x)^6+K[1]^2}dK[1]=c_1,y(x)\right ] \]

✓ Sympy. Time used: 4.149 (sec). Leaf size: 31

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

\[ C_{1} + \frac {\log {\left (x \right )}}{3} + \frac {\log {\left (1 + \frac {y^{6}{\left (x \right )}}{x^{2}} \right )}}{6} - \frac {\operatorname {atan}{\left (\frac {y^{3}{\left (x \right )}}{x} \right )}}{3} = 0 \]