Internal problem ID [7837]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 257.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {x \left (x y+x^{4}-1\right ) y^{\prime }-y \left (x y-x^{4}-1\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 98
dsolve(x*(x*y(x)+x^4-1)*diff(y(x),x)-y(x)*(x*y(x)-x^4-1)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (-c_{1}+{\mathrm e}^{\RootOf \left (-2 \textit {\_Z} \,x^{4} {\mathrm e}^{2 \textit {\_Z}}+2 x^{4} {\mathrm e}^{2 \textit {\_Z}}-2 \,{\mathrm e}^{\textit {\_Z}} c_{1} x^{4}+{\mathrm e}^{2 \textit {\_Z}}-2 c_{1} {\mathrm e}^{\textit {\_Z}}+c_{1}^{2}\right )}\right ) {\mathrm e}^{-\RootOf \left (-2 \textit {\_Z} \,x^{4} {\mathrm e}^{2 \textit {\_Z}}+2 x^{4} {\mathrm e}^{2 \textit {\_Z}}-2 \,{\mathrm e}^{\textit {\_Z}} c_{1} x^{4}+{\mathrm e}^{2 \textit {\_Z}}-2 c_{1} {\mathrm e}^{\textit {\_Z}}+c_{1}^{2}\right )}}{x} \]
✓ Solution by Mathematica
Time used: 0.281 (sec). Leaf size: 39
DSolve[x*(x*y[x]+x^4-1)*y'[x]-y[x]*(x*y[x]-x^4-1)==0,y[x],x,IncludeSingularSolutions -> True]
\[ \operatorname {Solve}\left [2 x^2+\frac {y(x)}{x}+\frac {x \left (-2 \log \left (\frac {1}{1-x y(x)}\right )-2+c_1\right )}{y(x)}=0,y(x)\right ] \]