Internal problem ID [8052]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 472.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]
Solve \begin {gather*} \boxed {\left (x +y\right ) \left (y^{\prime }\right )^{2}+2 x y^{\prime }-y=0} \end {gather*}
✓ Solution by Maple
Time used: 1.525 (sec). Leaf size: 138
dsolve((x+y(x))*diff(y(x),x)^2+2*x*diff(y(x),x)-y(x) = 0,y(x), singsol=all)
\begin{align*} \ln \relax (x )-\arctanh \left (\frac {2 x +y \relax (x )}{2 x \sqrt {\frac {x^{2}+x y \relax (x )+y \relax (x )^{2}}{x^{2}}}}\right )+\ln \left (\frac {y \relax (x )}{x}\right )-c_{1} = 0 \\ \ln \relax (x )+\arctanh \left (\frac {2 x +y \relax (x )}{2 x \sqrt {\frac {x^{2}+x y \relax (x )+y \relax (x )^{2}}{x^{2}}}}\right )+\ln \left (\frac {y \relax (x )}{x}\right )-c_{1} = 0 \\ y \relax (x ) = \frac {\sqrt {3}\, x \tan \left (\RootOf \left (\sqrt {3}\, \ln \left (\frac {3 x^{2}}{4}+\frac {3 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}}{4}\right )+2 \sqrt {3}\, c_{1}+2 \textit {\_Z} \right )\right )}{2}-\frac {x}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.888 (sec). Leaf size: 166
DSolve[-y[x] + 2*x*y'[x] + (x + y[x])*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {2}{3} \sqrt {e^{c_1} \left (-3 x+e^{c_1}\right )}-\frac {e^{c_1}}{3} \\ y(x)\to \frac {2}{3} \sqrt {e^{c_1} \left (-3 x+e^{c_1}\right )}-\frac {e^{c_1}}{3} \\ y(x)\to e^{c_1}-2 \sqrt {e^{c_1} \left (x+e^{c_1}\right )} \\ y(x)\to 2 \sqrt {e^{c_1} \left (x+e^{c_1}\right )}+e^{c_1} \\ y(x)\to 0 \\ y(x)\to -\frac {1}{2} i \left (\sqrt {3}-i\right ) x \\ y(x)\to \frac {1}{2} i \left (\sqrt {3}+i\right ) x \\ \end{align*}