23.1.421 problem 410

Internal problem ID [5028]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 410
Date solved : Tuesday, September 30, 2025 at 11:27:31 AM
CAS classification : [_separable]

\begin{align*} y^{\prime } \left (x^{3}+1\right )^{{2}/{3}}+\left (1+y^{3}\right )^{{2}/{3}}&=0 \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 119

ode:=diff(y(x),x)*(x^3+1)^(2/3)+(1+y(x)^3)^(2/3) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ c_1 +\frac {2 \pi \sqrt {3}\, \left (\left (-y^{3}\right )^{{1}/{6}} \operatorname {LegendreP}\left (-\frac {1}{3}, -\frac {1}{3}, \frac {-x^{3}+1}{x^{3}+1}\right ) \left (1+y^{3}\right )^{{1}/{3}} x +\operatorname {LegendreP}\left (-\frac {1}{3}, -\frac {1}{3}, \frac {-y^{3}+1}{1+y^{3}}\right ) \left (-x^{3}\right )^{{1}/{6}} y \left (x^{3}+1\right )^{{1}/{3}}\right )}{9 \left (-x^{3}\right )^{{1}/{6}} \left (x^{3}+1\right )^{{1}/{3}} \left (-y^{3}\right )^{{1}/{6}} \left (1+y^{3}\right )^{{1}/{3}} \Gamma \left (\frac {2}{3}\right )} = 0 \]

✓ Mathematica. Time used: 0.28 (sec). Leaf size: 72

ode=D[y[x],x]*(1+x^3)^(2/3)+(1+y[x]^3)^(2/3)==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {InverseFunction}\left [\text {$\#$1} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\text {$\#$1}^3\right )\&\right ]\left [-x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-x^3\right )+c_1\right ]\\ y(x)&\to -1\\ y(x)&\to \sqrt [3]{-1}\\ y(x)&\to -(-1)^{2/3} \end{align*}

✓ Sympy. Time used: 0.452 (sec). Leaf size: 61

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x**3 + 1)**(2/3)*Derivative(y(x), x) + (y(x)**3 + 1)**(2/3),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \frac {y{\left (x \right )} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {2}{3} \\ \frac {4}{3} \end {matrix}\middle | {e^{i \pi } y^{3}{\left (x \right )}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} = C_{1} - \frac {x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {2}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} \]