Internal problem ID [3802]
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`]]
\[ \boxed {x \left (1+x -2 y\right ) y^{\prime }+\left (1-2 x +y\right ) y=0} \]
✓ 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 \left (x \right ) = \frac {3 \,5^{\frac {1}{3}} {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x +80 c_{1} -x}{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 +80 c_{1} -x}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}-x -1 y \left (x \right ) = -\frac {3 \,5^{\frac {1}{3}} {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x +80 c_{1} -x}{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 +80 c_{1} -x}{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 +80 c_{1} -x}{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 +80 c_{1} -x}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}\right )}{2} y \left (x \right ) = -\frac {3 \,5^{\frac {1}{3}} {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x +80 c_{1} -x}{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 +80 c_{1} -x}{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 +80 c_{1} -x}{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 +80 c_{1} -x}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}\right )}{2} \end{align*}
✓ Solution by Mathematica
Time used: 44.02 (sec). Leaf size: 471
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]{2} x}{\sqrt [3]{27 c_1{}^2 x^2+\sqrt {\left (27 c_1{}^2 x^2+27 c_1{}^2 x\right ){}^2-108 c_1{}^3 x^3}+27 c_1{}^2 x}}-\frac {\sqrt [3]{27 c_1{}^2 x^2+\sqrt {\left (27 c_1{}^2 x^2+27 c_1{}^2 x\right ){}^2-108 c_1{}^3 x^3}+27 c_1{}^2 x}}{3 \sqrt [3]{2} c_1}-x-1 y(x)\to \frac {\left (1+i \sqrt {3}\right ) x}{2^{2/3} \sqrt [3]{27 c_1{}^2 x^2+\sqrt {\left (27 c_1{}^2 x^2+27 c_1{}^2 x\right ){}^2-108 c_1{}^3 x^3}+27 c_1{}^2 x}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{27 c_1{}^2 x^2+\sqrt {\left (27 c_1{}^2 x^2+27 c_1{}^2 x\right ){}^2-108 c_1{}^3 x^3}+27 c_1{}^2 x}}{6 \sqrt [3]{2} c_1}-x-1 y(x)\to \frac {\left (1-i \sqrt {3}\right ) x}{2^{2/3} \sqrt [3]{27 c_1{}^2 x^2+\sqrt {\left (27 c_1{}^2 x^2+27 c_1{}^2 x\right ){}^2-108 c_1{}^3 x^3}+27 c_1{}^2 x}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{27 c_1{}^2 x^2+\sqrt {\left (27 c_1{}^2 x^2+27 c_1{}^2 x\right ){}^2-108 c_1{}^3 x^3}+27 c_1{}^2 x}}{6 \sqrt [3]{2} c_1}-x-1 y(x)\to \text {Indeterminate} y(x)\to -x-1 \end{align*}