29.28.14 problem 812

Internal problem ID [5395]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 28
Problem number : 812
Date solved : Monday, January 27, 2025 at 11:18:49 AM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{2}+\left (1+2 y\right ) y^{\prime }+y \left (y-1\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.036 (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 \left (x \right )-1\right )}{2}-\frac {\ln \left (y \left (x \right )\right )}{2}+\frac {\ln \left (\sqrt {8 y \left (x \right )+1}-1\right )}{2}+\frac {3 \ln \left (\sqrt {8 y \left (x \right )+1}+3\right )}{2}-\frac {\ln \left (\sqrt {8 y \left (x \right )+1}+1\right )}{2}-\frac {3 \ln \left (\sqrt {8 y \left (x \right )+1}-3\right )}{2}-c_{1} &= 0 \\ x +\frac {3 \ln \left (y \left (x \right )-1\right )}{2}-\frac {\ln \left (y \left (x \right )\right )}{2}-\frac {\ln \left (\sqrt {8 y \left (x \right )+1}-1\right )}{2}-\frac {3 \ln \left (\sqrt {8 y \left (x \right )+1}+3\right )}{2}+\frac {\ln \left (\sqrt {8 y \left (x \right )+1}+1\right )}{2}+\frac {3 \ln \left (\sqrt {8 y \left (x \right )+1}-3\right )}{2}-c_{1} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.114 (sec). Leaf size: 1373

DSolve[(D[y[x],x])^2+(1+2*y[x])*D[y[x],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]{24 \sqrt {3} \sqrt {e^{7 (x+2 c_1)} \left (-1+27 e^{x+2 c_1}\right ){}^3}+5832 e^{5 (x+2 c_1)}-e^{3 x+6 c_1}+540 e^{4 x+8 c_1}}+\frac {64 e^{2 x+4 c_1} \left (1+216 e^{x+2 c_1}\right )}{\sqrt [3]{24 \sqrt {3} \sqrt {e^{7 (x+2 c_1)} \left (-1+27 e^{x+2 c_1}\right ){}^3}+5832 e^{5 (x+2 c_1)}-e^{3 x+6 c_1}+540 e^{4 x+8 c_1}}}\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]{24 \sqrt {3} \sqrt {e^{7 (x+2 c_1)} \left (-1+27 e^{x+2 c_1}\right ){}^3}+5832 e^{5 (x+2 c_1)}-e^{3 x+6 c_1}+540 e^{4 x+8 c_1}}-\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]{24 \sqrt {3} \sqrt {e^{7 (x+2 c_1)} \left (-1+27 e^{x+2 c_1}\right ){}^3}+5832 e^{5 (x+2 c_1)}-e^{3 x+6 c_1}+540 e^{4 x+8 c_1}}}\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]{24 \sqrt {3} \sqrt {e^{7 (x+2 c_1)} \left (-1+27 e^{x+2 c_1}\right ){}^3}+5832 e^{5 (x+2 c_1)}-e^{3 x+6 c_1}+540 e^{4 x+8 c_1}}+\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]{24 \sqrt {3} \sqrt {e^{7 (x+2 c_1)} \left (-1+27 e^{x+2 c_1}\right ){}^3}+5832 e^{5 (x+2 c_1)}-e^{3 x+6 c_1}+540 e^{4 x+8 c_1}}}\right )}{3072} \\ \end{align*}