40.2.7 problem 30

Internal problem ID [6585]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 4. Equations of first order and first degree (Variable separable). Supplemetary problems. Page 22
Problem number : 30
Date solved : Wednesday, March 05, 2025 at 12:58:41 AM
CAS classification : [[_homogeneous, `class D`], _rational]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 655

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

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 44.01 (sec). Leaf size: 423

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

\begin{align*} y(x)\to \frac {\sqrt [3]{2} \left (2 x^2 \log (x)+(-3+2 c_1) x^2-2\right )}{\sqrt [3]{54 x^3+2 \sqrt {729 x^6+\left (-6 x^2 \log (x)+(9-6 c_1) x^2+6\right ){}^3}}}+\frac {\sqrt [3]{54 x^3+2 \sqrt {729 x^6+\left (-6 x^2 \log (x)+(9-6 c_1) x^2+6\right ){}^3}}}{3 \sqrt [3]{2}} \\ y(x)\to \frac {2 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) \left (-6 x^2 \log (x)+(9-6 c_1) x^2+6\right )+2^{2/3} \left (-1+i \sqrt {3}\right ) \left (54 x^3+2 \sqrt {729 x^6+\left (-6 x^2 \log (x)+(9-6 c_1) x^2+6\right ){}^3}\right ){}^{2/3}}{12 \sqrt [3]{54 x^3+2 \sqrt {729 x^6+\left (-6 x^2 \log (x)+(9-6 c_1) x^2+6\right ){}^3}}} \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) \left (2 x^2 \log (x)+(-3+2 c_1) x^2-2\right )}{2^{2/3} \sqrt [3]{54 x^3+2 \sqrt {729 x^6+\left (-6 x^2 \log (x)+(9-6 c_1) x^2+6\right ){}^3}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{54 x^3+2 \sqrt {729 x^6+\left (-6 x^2 \log (x)+(9-6 c_1) x^2+6\right ){}^3}}}{6 \sqrt [3]{2}} \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (x**3 + x**2*y(x) + y(x)**3 + 2*y(x))*y(x)/(x*(x**3 + y(x)**3)) cannot be solved by the factorable group method