60.2.260 problem 836

Internal problem ID [10847]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 836
Date solved : Monday, January 27, 2025 at 10:13:47 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class C`]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 79

dsolve(diff(y(x),x) = y(x)*(x-y(x))*(y(x)+1)/x/(x*y(x)+x-y(x)),y(x), singsol=all)
 

\[ y = -\frac {{\mathrm e}^{\operatorname {RootOf}\left ({\mathrm e}^{\textit {\_Z}} \ln \left (2\right )-{\mathrm e}^{\textit {\_Z}} \ln \left ({\mathrm e}^{\textit {\_Z}}+9\right )+3 c_{1} {\mathrm e}^{\textit {\_Z}}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}-x \,{\mathrm e}^{\textit {\_Z}}+9\right )} x}{-9+\left (x -1\right ) {\mathrm e}^{\operatorname {RootOf}\left ({\mathrm e}^{\textit {\_Z}} \ln \left (2\right )-{\mathrm e}^{\textit {\_Z}} \ln \left ({\mathrm e}^{\textit {\_Z}}+9\right )+3 c_{1} {\mathrm e}^{\textit {\_Z}}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}-x \,{\mathrm e}^{\textit {\_Z}}+9\right )}} \]

✓ Solution by Mathematica

Time used: 1.375 (sec). Leaf size: 105

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

\[ \text {Solve}\left [\int _1^{\frac {(x-1)^2 \left (\frac {x^6}{(x-1)^3}\right )^{2/3} (x+(x+2) y(x))}{\sqrt [3]{2} x^4 (x+(x-1) y(x))}}\frac {1}{K[1]^3-\frac {3 K[1]}{2^{2/3}}+1}dK[1]+\frac {2^{2/3} \left (\frac {x^6}{(x-1)^3}\right )^{2/3} (x-1)^2}{9 x^3}=c_1,y(x)\right ] \]