Internal problem ID [3661]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 14
Problem number: 407.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
\[ \boxed {y^{\prime } \left (x^{3}+1\right )^{\frac {2}{3}}+\left (1+y^{3}\right )^{\frac {2}{3}}=0} \]
✓ Solution by Maple
Time used: 0.031 (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)
\[ c_{1} +\frac {2 \pi \sqrt {3}\, \left (y \left (x \right ) \left (x^{3}+1\right )^{\frac {1}{3}} \left (-x^{3}\right )^{\frac {1}{6}} \operatorname {LegendreP}\left (-\frac {1}{3}, -\frac {1}{3}, \frac {-y \left (x \right )^{3}+1}{1+y \left (x \right )^{3}}\right )+\left (1+y \left (x \right )^{3}\right )^{\frac {1}{3}} \operatorname {LegendreP}\left (-\frac {1}{3}, -\frac {1}{3}, \frac {-x^{3}+1}{x^{3}+1}\right ) \left (-y \left (x \right )^{3}\right )^{\frac {1}{6}} x \right )}{9 \left (-x^{3}\right )^{\frac {1}{6}} \left (x^{3}+1\right )^{\frac {1}{3}} \left (-y \left (x \right )^{3}\right )^{\frac {1}{6}} \left (1+y \left (x \right )^{3}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}\right )} = 0 \]
✓ Solution by Mathematica
Time used: 3.082 (sec). Leaf size: 221
DSolve[y'[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 [\frac {3 \sqrt [3]{\frac {\sqrt [3]{-1}-\text {$\#$1}}{1+\sqrt [3]{-1}}} (\text {$\#$1}+1) \left (\frac {\text {$\#$1}+(-1)^{2/3}}{(-1)^{2/3}-1}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {\sqrt [3]{-1} (\text {$\#$1}+1)}{\left (-1+\sqrt [3]{-1}\right ) \text {$\#$1}+1}\right )}{\left (\text {$\#$1}^3+1\right )^{2/3}}\&\right ]\left [-\frac {3 \sqrt [3]{\frac {\sqrt [3]{-1}-x}{1+\sqrt [3]{-1}}} (x+1) \left (\frac {x+(-1)^{2/3}}{(-1)^{2/3}-1}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {\sqrt [3]{-1} (x+1)}{\left (-1+\sqrt [3]{-1}\right ) x+1}\right )}{\left (x^3+1\right )^{2/3}}+c_1\right ] \\ y(x)\to -1 \\ y(x)\to \sqrt [3]{-1} \\ y(x)\to -(-1)^{2/3} \\ \end{align*}