12.4.10 problem 10

Internal problem ID [1617]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Existence and Uniqueness of Solutions of Nonlinear Equations. Section 2.3 Page 60
Problem number : 10
Date solved : Saturday, March 29, 2025 at 11:07:51 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 44

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

\[ \frac {x^{2}}{2}-\frac {{\left (-\operatorname {signum}\left (y^{2}-1\right )\right )}^{{2}/{3}} y \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {2}{3}\right ], \left [\frac {3}{2}\right ], y^{2}\right )}{\operatorname {signum}\left (y^{2}-1\right )^{{2}/{3}}}+c_1 = 0 \]

✓ Mathematica. Time used: 0.339 (sec). Leaf size: 66

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

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

✓ Sympy. Time used: 0.498 (sec). Leaf size: 29

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

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