23.1.416 problem 405

Internal problem ID [5023]
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 : 405
Date solved : Tuesday, September 30, 2025 at 11:27:15 AM
CAS classification : [_separable]

\begin{align*} y^{\prime } \sqrt {-x^{4}+1}&=\sqrt {1-y^{4}} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 32

ode:=diff(y(x),x)*(-x^4+1)^(1/2) = (1-y(x)^4)^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ \int \frac {1}{\sqrt {-x^{4}+1}}d x -\int _{}^{y}\frac {1}{\sqrt {-\textit {\_a}^{4}+1}}d \textit {\_a} +c_1 = 0 \]

✓ Mathematica. Time used: 0.191 (sec). Leaf size: 38

ode=D[y[x],x]*Sqrt[1-x^4]==Sqrt[1-y[x]^4]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {sn}(c_1+\operatorname {EllipticF}(\arcsin (x),-1)|-1)\\ y(x)&\to -1\\ y(x)&\to -i\\ y(x)&\to i\\ y(x)&\to 1 \end{align*}

✓ Sympy. Time used: 0.463 (sec). Leaf size: 65

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(sqrt(1 - x**4)*Derivative(y(x), x) - sqrt(1 - y(x)**4),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \frac {y{\left (x \right )} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {e^{2 i \pi } y^{4}{\left (x \right )}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} = C_{1} + \frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \]