Internal problem ID [3351]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 22
Problem number: 609.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {y \left (1+y\right ) y^{\prime }-x \left (x +1\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 720
dsolve(y(x)*(1+y(x))*diff(y(x),x) = x*(1+x),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (-1+4 x^{3}+6 x^{2}+12 c_{1}+2 \sqrt {4 x^{6}+12 x^{5}+24 x^{3} c_{1}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{\frac {1}{3}}}{2}+\frac {1}{2 \left (-1+4 x^{3}+6 x^{2}+12 c_{1}+2 \sqrt {4 x^{6}+12 x^{5}+24 x^{3} c_{1}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{\frac {1}{3}}}-\frac {1}{2} \\ y \relax (x ) = -\frac {\left (-1+4 x^{3}+6 x^{2}+12 c_{1}+2 \sqrt {4 x^{6}+12 x^{5}+24 x^{3} c_{1}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{\frac {1}{3}}}{4}-\frac {1}{4 \left (-1+4 x^{3}+6 x^{2}+12 c_{1}+2 \sqrt {4 x^{6}+12 x^{5}+24 x^{3} c_{1}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{\frac {1}{3}}}-\frac {1}{2}-\frac {i \sqrt {3}\, \left (\frac {\left (-1+4 x^{3}+6 x^{2}+12 c_{1}+2 \sqrt {4 x^{6}+12 x^{5}+24 x^{3} c_{1}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{\frac {1}{3}}}{2}-\frac {1}{2 \left (-1+4 x^{3}+6 x^{2}+12 c_{1}+2 \sqrt {4 x^{6}+12 x^{5}+24 x^{3} c_{1}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (-1+4 x^{3}+6 x^{2}+12 c_{1}+2 \sqrt {4 x^{6}+12 x^{5}+24 x^{3} c_{1}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{\frac {1}{3}}}{4}-\frac {1}{4 \left (-1+4 x^{3}+6 x^{2}+12 c_{1}+2 \sqrt {4 x^{6}+12 x^{5}+24 x^{3} c_{1}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{\frac {1}{3}}}-\frac {1}{2}+\frac {i \sqrt {3}\, \left (\frac {\left (-1+4 x^{3}+6 x^{2}+12 c_{1}+2 \sqrt {4 x^{6}+12 x^{5}+24 x^{3} c_{1}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{\frac {1}{3}}}{2}-\frac {1}{2 \left (-1+4 x^{3}+6 x^{2}+12 c_{1}+2 \sqrt {4 x^{6}+12 x^{5}+24 x^{3} c_{1}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.229 (sec). Leaf size: 346
DSolve[y[x](1+y[x])y'[x]==x(1+x),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \left (\sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}+\frac {1}{\sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}}-1\right ) \\ y(x)\to \frac {1}{8} \left (2 i \left (\sqrt {3}+i\right ) \sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}+\frac {-2-2 i \sqrt {3}}{\sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}}-4\right ) \\ y(x)\to \frac {1}{8} \left (-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}+\frac {2 i \left (\sqrt {3}+i\right )}{\sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}}-4\right ) \\ \end{align*}