23.1.414 problem 403

Internal problem ID [5021]
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 : 403
Date solved : Tuesday, September 30, 2025 at 11:27:08 AM
CAS classification : [_separable]

\begin{align*} y^{\prime } \sqrt {x^{3}+1}&=\sqrt {1+y^{3}} \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 28

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

\[ \int \frac {1}{\sqrt {x^{3}+1}}d x -\int _{}^{y}\frac {1}{\sqrt {\textit {\_a}^{3}+1}}d \textit {\_a} +c_1 = 0 \]

✓ Mathematica. Time used: 0.241 (sec). Leaf size: 71

ode=D[y[x],x]*Sqrt[1+x^3]==Sqrt[1+y[x]^3]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {InverseFunction}\left [\text {$\#$1} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\text {$\#$1}^3\right )\&\right ]\left [x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\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: 0.449 (sec). Leaf size: 61

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

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