Internal problem ID [3543]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 28
Problem number: 812.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}+\left (1+2 y\right ) y^{\prime }+y \left (y-1\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 6.328 (sec). Leaf size: 143
dsolve(diff(y(x),x)^2+(1+2*y(x))*diff(y(x),x)+y(x)*(y(x)-1) = 0,y(x), singsol=all)
\begin{align*} x +\frac {3 \ln \left (y \relax (x )-1\right )}{2}-\frac {\ln \left (y \relax (x )\right )}{2}+\frac {3 \ln \left (\sqrt {8 y \relax (x )+1}+3\right )}{2}+\frac {\ln \left (\sqrt {8 y \relax (x )+1}-1\right )}{2}-\frac {3 \ln \left (\sqrt {8 y \relax (x )+1}-3\right )}{2}-\frac {\ln \left (\sqrt {8 y \relax (x )+1}+1\right )}{2}-c_{1} = 0 \\ x +\frac {3 \ln \left (y \relax (x )-1\right )}{2}-\frac {\ln \left (y \relax (x )\right )}{2}-\frac {3 \ln \left (\sqrt {8 y \relax (x )+1}+3\right )}{2}-\frac {\ln \left (\sqrt {8 y \relax (x )+1}-1\right )}{2}+\frac {3 \ln \left (\sqrt {8 y \relax (x )+1}-3\right )}{2}+\frac {\ln \left (\sqrt {8 y \relax (x )+1}+1\right )}{2}-c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.113 (sec). Leaf size: 1367
DSolve[(y'[x])^2+(1+2 y[x])y'[x]+y[x](y[x]-1)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{-2 x} \left (128 e^x \left (12 e^x+e^{2 c_1}\right )+64 \sqrt [3]{24 \sqrt {3} \sqrt {-e^{7 x+4 c_1} \left (-27 e^x+e^{2 c_1}\right ){}^3}+540 e^{4 (x+c_1)}+5832 e^{5 x+2 c_1}-e^{3 x+6 c_1}}+\frac {64 e^{2 (x+c_1)} \left (216 e^x+e^{2 c_1}\right )}{\sqrt [3]{24 \sqrt {3} \sqrt {-e^{7 x+4 c_1} \left (-27 e^x+e^{2 c_1}\right ){}^3}+540 e^{4 (x+c_1)}+5832 e^{5 x+2 c_1}-e^{3 x+6 c_1}}}\right )}{1536} \\ y(x)\to \frac {e^{-2 x} \left (256 e^x \left (12 e^x+e^{2 c_1}\right )+64 i \left (\sqrt {3}+i\right ) \sqrt [3]{24 \sqrt {3} \sqrt {-e^{7 x+4 c_1} \left (-27 e^x+e^{2 c_1}\right ){}^3}+540 e^{4 (x+c_1)}+5832 e^{5 x+2 c_1}-e^{3 x+6 c_1}}-\frac {64 i \left (\sqrt {3}-i\right ) e^{2 (x+c_1)} \left (216 e^x+e^{2 c_1}\right )}{\sqrt [3]{24 \sqrt {3} \sqrt {-e^{7 x+4 c_1} \left (-27 e^x+e^{2 c_1}\right ){}^3}+540 e^{4 (x+c_1)}+5832 e^{5 x+2 c_1}-e^{3 x+6 c_1}}}\right )}{3072} \\ y(x)\to \frac {e^{-2 x} \left (256 e^x \left (12 e^x+e^{2 c_1}\right )-64 \left (1+i \sqrt {3}\right ) \sqrt [3]{24 \sqrt {3} \sqrt {-e^{7 x+4 c_1} \left (-27 e^x+e^{2 c_1}\right ){}^3}+540 e^{4 (x+c_1)}+5832 e^{5 x+2 c_1}-e^{3 x+6 c_1}}+\frac {64 i \left (\sqrt {3}+i\right ) e^{2 (x+c_1)} \left (216 e^x+e^{2 c_1}\right )}{\sqrt [3]{24 \sqrt {3} \sqrt {-e^{7 x+4 c_1} \left (-27 e^x+e^{2 c_1}\right ){}^3}+540 e^{4 (x+c_1)}+5832 e^{5 x+2 c_1}-e^{3 x+6 c_1}}}\right )}{3072} \\ y(x)\to \frac {e^{-2 (x+2 c_1)} \left (128 e^{x+2 c_1} \left (1+12 e^{x+2 c_1}\right )+64 \sqrt [3]{e^{3 x+6 c_1} \left (-1+108 e^{x+2 c_1} \left (5+54 e^{x+2 c_1}\right )\right )+24 \sqrt {3} \sqrt {e^{7 (x+2 c_1)} \left (-1+27 e^{x+2 c_1}\right ){}^3}}+\frac {64 e^{2 x+4 c_1} \left (1+216 e^{x+2 c_1}\right )}{\sqrt [3]{e^{3 x+6 c_1} \left (-1+108 e^{x+2 c_1} \left (5+54 e^{x+2 c_1}\right )\right )+24 \sqrt {3} \sqrt {e^{7 (x+2 c_1)} \left (-1+27 e^{x+2 c_1}\right ){}^3}}}\right )}{1536} \\ y(x)\to \frac {e^{-2 (x+2 c_1)} \left (256 e^{x+2 c_1} \left (1+12 e^{x+2 c_1}\right )+64 i \left (\sqrt {3}+i\right ) \sqrt [3]{e^{3 x+6 c_1} \left (-1+108 e^{x+2 c_1} \left (5+54 e^{x+2 c_1}\right )\right )+24 \sqrt {3} \sqrt {e^{7 (x+2 c_1)} \left (-1+27 e^{x+2 c_1}\right ){}^3}}-\frac {64 i \left (\sqrt {3}-i\right ) e^{2 x+4 c_1} \left (1+216 e^{x+2 c_1}\right )}{\sqrt [3]{e^{3 x+6 c_1} \left (-1+108 e^{x+2 c_1} \left (5+54 e^{x+2 c_1}\right )\right )+24 \sqrt {3} \sqrt {e^{7 (x+2 c_1)} \left (-1+27 e^{x+2 c_1}\right ){}^3}}}\right )}{3072} \\ y(x)\to \frac {e^{-2 (x+2 c_1)} \left (256 e^{x+2 c_1} \left (1+12 e^{x+2 c_1}\right )-64 \left (1+i \sqrt {3}\right ) \sqrt [3]{e^{3 x+6 c_1} \left (-1+108 e^{x+2 c_1} \left (5+54 e^{x+2 c_1}\right )\right )+24 \sqrt {3} \sqrt {e^{7 (x+2 c_1)} \left (-1+27 e^{x+2 c_1}\right ){}^3}}+\frac {64 i \left (\sqrt {3}+i\right ) e^{2 x+4 c_1} \left (1+216 e^{x+2 c_1}\right )}{\sqrt [3]{e^{3 x+6 c_1} \left (-1+108 e^{x+2 c_1} \left (5+54 e^{x+2 c_1}\right )\right )+24 \sqrt {3} \sqrt {e^{7 (x+2 c_1)} \left (-1+27 e^{x+2 c_1}\right ){}^3}}}\right )}{3072} \\ \end{align*}