Internal problem ID [3294]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 20
Problem number: 550.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {x \left (1+x -2 y\right ) y^{\prime }+\left (1-2 x +y\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 499
dsolve(x*(1+x-2*y(x))*diff(y(x),x)+(1-2*x+y(x))*y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {3 \,5^{\frac {1}{3}} \left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x -x +80 c_{1}}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )^{\frac {1}{3}}}{40 c_{1}}+\frac {3 x 5^{\frac {2}{3}}}{40 \left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x -x +80 c_{1}}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )^{\frac {1}{3}}}-x -1 \\ y \relax (x ) = -\frac {3 \,5^{\frac {1}{3}} \left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x -x +80 c_{1}}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )^{\frac {1}{3}}}{80 c_{1}}-\frac {3 x 5^{\frac {2}{3}}}{80 \left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x -x +80 c_{1}}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )^{\frac {1}{3}}}-x -1-\frac {i \sqrt {3}\, \left (\frac {3 \,5^{\frac {1}{3}} \left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x -x +80 c_{1}}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )^{\frac {1}{3}}}{40 c_{1}}-\frac {3 x 5^{\frac {2}{3}}}{40 \left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x -x +80 c_{1}}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {3 \,5^{\frac {1}{3}} \left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x -x +80 c_{1}}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )^{\frac {1}{3}}}{80 c_{1}}-\frac {3 x 5^{\frac {2}{3}}}{80 \left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x -x +80 c_{1}}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )^{\frac {1}{3}}}-x -1+\frac {i \sqrt {3}\, \left (\frac {3 \,5^{\frac {1}{3}} \left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x -x +80 c_{1}}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )^{\frac {1}{3}}}{40 c_{1}}-\frac {3 x 5^{\frac {2}{3}}}{40 \left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x -x +80 c_{1}}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 37.224 (sec). Leaf size: 448
DSolve[x(1+x-2 y[x])y'[x]+(1-2 x+y[x])y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt [3]{\frac {2}{3}} x}{\sqrt [3]{\sqrt {3} \sqrt {c_1{}^3 x^2 \left (-4 x+27 c_1 (x+1)^2\right )}+9 c_1{}^2 x (x+1)}}-\frac {\sqrt [3]{\sqrt {3} \sqrt {c_1{}^3 x^2 \left (-4 x+27 c_1 (x+1)^2\right )}+9 c_1{}^2 x (x+1)}}{\sqrt [3]{2} 3^{2/3} c_1}-x-1 \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {3} \sqrt {c_1{}^3 x^2 \left (-4 x+27 c_1 (x+1)^2\right )}+9 c_1{}^2 x (x+1)}}{2 \sqrt [3]{2} 3^{2/3} c_1}+\frac {x+i \sqrt {3} x}{2^{2/3} \sqrt [3]{3} \sqrt [3]{\sqrt {3} \sqrt {c_1{}^3 x^2 \left (-4 x+27 c_1 (x+1)^2\right )}+9 c_1{}^2 x (x+1)}}-x-1 \\ y(x)\to \frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {3} \sqrt {c_1{}^3 x^2 \left (-4 x+27 c_1 (x+1)^2\right )}+9 c_1{}^2 x (x+1)}}{2 \sqrt [3]{2} 3^{2/3} c_1}+\frac {x-i \sqrt {3} x}{2^{2/3} \sqrt [3]{3} \sqrt [3]{\sqrt {3} \sqrt {c_1{}^3 x^2 \left (-4 x+27 c_1 (x+1)^2\right )}+9 c_1{}^2 x (x+1)}}-x-1 \\ y(x)\to \text {Indeterminate} \\ y(x)\to -x-1 \\ \end{align*}