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 : Monday, January 27, 2025 at 05:14:08 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 44

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

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

✓ Solution by Mathematica

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

DSolve[D[y[x],x]==x*(y[x]^2-1)^(2/3),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*}