Internal problem ID [8551]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 971.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_1st_order, _with_symmetry_[F(x),G(x)]], _Abel]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {\left (y x +1\right )^{3}}{x^{5}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 91
dsolve(diff(y(x),x) = (x*y(x)+1)^3/x^5,y(x), singsol=all)
\[ y \relax (x ) = \frac {\sqrt {3}\, \left (-\frac {1}{x^{6}}\right )^{\frac {1}{3}} x^{3}+3 \tan \left (\RootOf \left (-18 x^{3} \left (-\frac {1}{x^{6}}\right )^{\frac {2}{3}}-6 \textit {\_Z} \sqrt {3}-\ln \left (\frac {27 \left (\sqrt {3}+\tan \left (\textit {\_Z} \right )\right )^{6}}{\left (\left (\sqrt {3}\right )^{2}+3 \left (\tan ^{2}\left (\textit {\_Z} \right )\right )\right )^{3}}\right )+18 c_{1}\right )\right ) x^{3} \left (-\frac {1}{x^{6}}\right )^{\frac {1}{3}}-2 \sqrt {3}}{2 \sqrt {3}\, x} \]
✓ Solution by Mathematica
Time used: 0.212 (sec). Leaf size: 157
DSolve[y'[x] == (1 + x*y[x])^3/x^5,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {\text {ArcTan}\left (\frac {\frac {2 \left (\frac {3}{x^3}+\frac {3 y(x)}{x^2}\right )}{3 \sqrt [3]{-\frac {1}{x^6}}}-1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (\frac {\frac {3}{x^3}+\frac {3 y(x)}{x^2}}{3 \sqrt [3]{-\frac {1}{x^6}}}+1\right )-\frac {1}{6} \log \left (\frac {\left (\frac {3}{x^3}+\frac {3 y(x)}{x^2}\right )^2}{9 \left (-\frac {1}{x^6}\right )^{2/3}}-\frac {\frac {3}{x^3}+\frac {3 y(x)}{x^2}}{3 \sqrt [3]{-\frac {1}{x^6}}}+1\right )=-\left (-\frac {1}{x^6}\right )^{2/3} x^3+c_1,y(x)\right ] \]