Internal problem ID [7646]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 65.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {y^{\prime }-\sqrt {\frac {y^{3}+1}{x^{3}+1}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 47
dsolve(diff(y(x),x) - sqrt((y(x)^3+1)/(x^3+1))=0,y(x), singsol=all)
\[ \int _{}^{y \relax (x )}\frac {1}{\sqrt {\textit {\_a}^{3}+1}}d \textit {\_a} +\int _{}^{x}-\frac {\sqrt {\frac {y \relax (x )^{3}+1}{\textit {\_a}^{3}+1}}}{\sqrt {y \relax (x )^{3}+1}}d \textit {\_a} +c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 99.853 (sec). Leaf size: 337
DSolve[y'[x] - Sqrt[(y[x]^3+1)/(x^3+1)]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {i (\text {$\#$1}+1) \sqrt {1+\frac {6 i}{\left (\sqrt {3}-3 i\right ) (\text {$\#$1}+1)}} \sqrt {\frac {2}{3}-\frac {4 i}{\left (\sqrt {3}+3 i\right ) (\text {$\#$1}+1)}} \text {EllipticF}\left (i \sinh ^{-1}\left (\frac {\sqrt {-\frac {6 i}{\sqrt {3}+3 i}}}{\sqrt {\text {$\#$1}+1}}\right ),\frac {\sqrt {3}+3 i}{-\sqrt {3}+3 i}\right )}{\sqrt {-\frac {i}{\sqrt {3}+3 i}} \sqrt {\text {$\#$1}^2-\text {$\#$1}+1}}\&\right ]\left [\frac {i (x+1) \sqrt {1+\frac {6 i}{\left (\sqrt {3}-3 i\right ) (x+1)}} \sqrt {\frac {2}{3}-\frac {4 i}{\left (\sqrt {3}+3 i\right ) (x+1)}} \text {EllipticF}\left (i \sinh ^{-1}\left (\frac {\sqrt {-\frac {6 i}{\sqrt {3}+3 i}}}{\sqrt {x+1}}\right ),\frac {\sqrt {3}+3 i}{-\sqrt {3}+3 i}\right )}{\sqrt {-\frac {i}{\sqrt {3}+3 i}} \sqrt {x^2-x+1}}+c_1\right ] \\ y(x)\to -1 \\ y(x)\to \sqrt [3]{-1} \\ y(x)\to -(-1)^{2/3} \\ \end{align*}