29.14.26 problem 407

Internal problem ID [5005]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 14
Problem number : 407
Date solved : Monday, January 27, 2025 at 10:02:40 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*}

✓ Solution by Maple

Time used: 0.039 (sec). Leaf size: 119

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

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

✓ Solution by Mathematica

Time used: 0.464 (sec). Leaf size: 72

DSolve[D[y[x],x](1+x^3)^(2/3)+(1+y[x]^3)^(2/3)==0,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*}