60.2.296 problem 873

Internal problem ID [10883]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 873
Date solved : Monday, January 27, 2025 at 10:20:30 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 54

dsolve(diff(y(x),x) = 1/x*(1+2*y(x))/(-2+x+x*y(x)^2+3*x*y(x)^3+2*x*y(x)+2*x*y(x)^4),y(x), singsol=all)
 

\begin{align*} y &= -{\frac {1}{2}} \\ y &= \frac {{\mathrm e}^{\operatorname {RootOf}\left (2 x \,{\mathrm e}^{4 \textit {\_Z}}-3 \,{\mathrm e}^{3 \textit {\_Z}} x -6 x \,{\mathrm e}^{2 \textit {\_Z}}+48 c_{1} x \,{\mathrm e}^{\textit {\_Z}}+54 \textit {\_Z} x \,{\mathrm e}^{\textit {\_Z}}+7 x \,{\mathrm e}^{\textit {\_Z}}+96\right )}}{2}-\frac {1}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.138 (sec). Leaf size: 60

DSolve[D[y[x],x] == (1 + 2*y[x])/(x*(-2 + x + 2*x*y[x] + x*y[x]^2 + 3*x*y[x]^3 + 2*x*y[x]^4)),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {1}{4} K[1]^2-\frac {K[1]}{8}-\frac {9}{16 (2 K[1]+1)}+\frac {1}{16}\right )dK[1]-\frac {1}{2 x (2 y(x)+1)}=c_1,y(x)\right ] \]