Internal problem ID [9163]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 828.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]
\[ \boxed {y^{\prime }-\frac {\left (1+2 y\right ) \left (y+1\right )}{x \left (-2 y-2+x y^{3}+2 y^{4} x \right )}=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 67
dsolve(diff(y(x),x) = 1/x*(1+2*y(x))*(y(x)+1)/(-2*y(x)-2+x*y(x)^3+2*x*y(x)^4),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -1 \\ y \left (x \right ) &= -{\frac {1}{2}} \\ y \left (x \right ) &= \frac {{\mathrm e}^{\operatorname {RootOf}\left (16 \,{\mathrm e}^{\textit {\_Z}} x \ln \left ({\mathrm e}^{\textit {\_Z}}+1\right )-16 \,{\mathrm e}^{\textit {\_Z}} x \ln \left (2\right )+8 c_{1} x \,{\mathrm e}^{\textit {\_Z}}-2 \textit {\_Z} x \,{\mathrm e}^{\textit {\_Z}}+x \,{\mathrm e}^{3 \textit {\_Z}}-8 x \,{\mathrm e}^{2 \textit {\_Z}}+7 x \,{\mathrm e}^{\textit {\_Z}}+16\right )}}{2}-\frac {1}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.375 (sec). Leaf size: 56
DSolve[y'[x] == ((1 + y[x])*(1 + 2*y[x]))/(x*(-2 - 2*y[x] + x*y[x]^3 + 2*x*y[x]^4)),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {1}{8} y(x)^2+\frac {3 y(x)}{8}-\frac {1}{2 x (2 y(x)+1)}-\frac {1}{2} \log (y(x)+1)+\frac {1}{16} \log (2 y(x)+1)=c_1,y(x)\right ] \]