60.2.252 problem 828

Internal problem ID [10839]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 828
Date solved : Monday, January 27, 2025 at 10:11:50 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

\begin{align*} y^{\prime }&=\frac {\left (1+2 y\right ) \left (y+1\right )}{x \left (-2 y-2+x y^{3}+2 x y^{4}\right )} \end{align*}

✓ Solution by Maple

Time used: 0.017 (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 &= -1 \\ y &= -{\frac {1}{2}} \\ y &= \frac {{\mathrm e}^{\operatorname {RootOf}\left (-16 \,{\mathrm e}^{\textit {\_Z}} x \ln \left (2\right )+16 \,{\mathrm e}^{\textit {\_Z}} x \ln \left ({\mathrm e}^{\textit {\_Z}}+1\right )+8 c_{1} x \,{\mathrm e}^{\textit {\_Z}}-2 \textit {\_Z} x \,{\mathrm e}^{\textit {\_Z}}-8 x \,{\mathrm e}^{2 \textit {\_Z}}+7 x \,{\mathrm e}^{\textit {\_Z}}+{\mathrm e}^{3 \textit {\_Z}} x +16\right )}}{2}-\frac {1}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.177 (sec). Leaf size: 62

DSolve[D[y[x],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]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {K[1]}{4}-\frac {1}{2 (K[1]+1)}+\frac {1}{8 (2 K[1]+1)}+\frac {3}{8}\right )dK[1]-\frac {1}{2 x (2 y(x)+1)}=c_1,y(x)\right ] \]