Internal problem ID [4610]
Book: Engineering Mathematics. By K. A. Stroud. 5th edition. Industrial press Inc. NY.
2001
Section: Program 24. First order differential equations. Further problems 24. page 1068
Problem number: 32.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class A], _rational, [_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {2 x y+y^{2}}{x^{2}+2 x y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.024 (sec). Leaf size: 380
dsolve(diff(y(x),x)=(2*x*y(x)+y(x)^2)/(x^2+2*x*y(x)),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {12^{\frac {1}{3}} \left (x \left (\sqrt {3}\, \sqrt {\frac {x \left (27 c_{1} x -4\right )}{c_{1}}}+9 x \right ) c_{1}^{2}\right )^{\frac {1}{3}}}{6 c_{1}}+\frac {x 12^{\frac {2}{3}}}{6 \left (x \left (\sqrt {3}\, \sqrt {\frac {x \left (27 c_{1} x -4\right )}{c_{1}}}+9 x \right ) c_{1}^{2}\right )^{\frac {1}{3}}}+x \\ y \relax (x ) = -\frac {12^{\frac {1}{3}} \left (x \left (\sqrt {3}\, \sqrt {\frac {x \left (27 c_{1} x -4\right )}{c_{1}}}+9 x \right ) c_{1}^{2}\right )^{\frac {1}{3}}}{12 c_{1}}-\frac {x 12^{\frac {2}{3}}}{12 \left (x \left (\sqrt {3}\, \sqrt {\frac {x \left (27 c_{1} x -4\right )}{c_{1}}}+9 x \right ) c_{1}^{2}\right )^{\frac {1}{3}}}+x -\frac {i \sqrt {3}\, \left (\frac {12^{\frac {1}{3}} \left (x \left (\sqrt {3}\, \sqrt {\frac {x \left (27 c_{1} x -4\right )}{c_{1}}}+9 x \right ) c_{1}^{2}\right )^{\frac {1}{3}}}{6 c_{1}}-\frac {x 12^{\frac {2}{3}}}{6 \left (x \left (\sqrt {3}\, \sqrt {\frac {x \left (27 c_{1} x -4\right )}{c_{1}}}+9 x \right ) c_{1}^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {12^{\frac {1}{3}} \left (x \left (\sqrt {3}\, \sqrt {\frac {x \left (27 c_{1} x -4\right )}{c_{1}}}+9 x \right ) c_{1}^{2}\right )^{\frac {1}{3}}}{12 c_{1}}-\frac {x 12^{\frac {2}{3}}}{12 \left (x \left (\sqrt {3}\, \sqrt {\frac {x \left (27 c_{1} x -4\right )}{c_{1}}}+9 x \right ) c_{1}^{2}\right )^{\frac {1}{3}}}+x +\frac {i \sqrt {3}\, \left (\frac {12^{\frac {1}{3}} \left (x \left (\sqrt {3}\, \sqrt {\frac {x \left (27 c_{1} x -4\right )}{c_{1}}}+9 x \right ) c_{1}^{2}\right )^{\frac {1}{3}}}{6 c_{1}}-\frac {x 12^{\frac {2}{3}}}{6 \left (x \left (\sqrt {3}\, \sqrt {\frac {x \left (27 c_{1} x -4\right )}{c_{1}}}+9 x \right ) c_{1}^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 17.326 (sec). Leaf size: 370
DSolve[y'[x]==(2*x*y[x]+y[x]^2)/(x^2+2*x*y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt [3]{\frac {2}{3}} e^{c_1} x}{\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (27 x+4 e^{c_1}\right )}-9 e^{c_1} x^2}}+\frac {\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (27 x+4 e^{c_1}\right )}-9 e^{c_1} x^2}}{\sqrt [3]{2} 3^{2/3}}+x \\ y(x)\to \sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (27 x+4 e^{c_1}\right )}-9 e^{c_1} x^2} \text {Root}\left [18 \text {$\#$1}^3-1\&,3\right ]+\frac {\sqrt [3]{-\frac {2}{3}} e^{c_1} x}{\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (27 x+4 e^{c_1}\right )}-9 e^{c_1} x^2}}+x \\ y(x)\to \sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (27 x+4 e^{c_1}\right )}-9 e^{c_1} x^2} \text {Root}\left [18 \text {$\#$1}^3-1\&,2\right ]+\frac {e^{c_1} x \text {Root}\left [3 \text {$\#$1}^3+2\&,2\right ]}{\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (27 x+4 e^{c_1}\right )}-9 e^{c_1} x^2}}+x \\ y(x)\to x \\ \end{align*}