Internal problem ID [8807]
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]
\[ \boxed {\left (x +y\right ) {y^{\prime }}^{2}+2 y^{\prime } x -y=0} \]
✓ Solution by Maple
Time used: 0.093 (sec). Leaf size: 119
dsolve((x+y(x))*diff(y(x),x)^2+2*x*diff(y(x),x)-y(x) = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = x \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) y \left (x \right ) = x \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \ln \left (x \right )-\operatorname {arctanh}\left (\frac {y \left (x \right )+2 x}{2 x \sqrt {\frac {y \left (x \right )^{2}+x y \left (x \right )+x^{2}}{x^{2}}}}\right )+\ln \left (\frac {y \left (x \right )}{x}\right )-c_{1} = 0 \ln \left (x \right )+\operatorname {arctanh}\left (\frac {y \left (x \right )+2 x}{2 x \sqrt {\frac {y \left (x \right )^{2}+x y \left (x \right )+x^{2}}{x^{2}}}}\right )+\ln \left (\frac {y \left (x \right )}{x}\right )-c_{1} = 0 \end{align*}
✓ Solution by Mathematica
Time used: 4.051 (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*}