Internal problem ID [8053]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 475.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, _dAlembert]
\[ \boxed {4 y {y^{\prime }}^{2}+2 y^{\prime } x -y=0} \]
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 73
dsolve(4*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 ) = -\frac {i x}{2} \\ y \left (x \right ) = \frac {i x}{2} \\ y \left (x \right ) = 0 \\ y \left (x \right ) = \sqrt {c_{1}^{2}-x c_{1}} \\ y \left (x \right ) = \sqrt {c_{1}^{2}+x c_{1}} \\ y \left (x \right ) = -\sqrt {c_{1}^{2}-x c_{1}} \\ y \left (x \right ) = -\sqrt {c_{1}^{2}+x c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.549 (sec). Leaf size: 140
DSolve[-y[x] + 2*x*y'[x] + 4*y[x]*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{2} e^{2 c_1} \sqrt {-2 x+e^{4 c_1}} \\ y(x)\to \frac {1}{2} e^{2 c_1} \sqrt {-2 x+e^{4 c_1}} \\ y(x)\to -\frac {1}{2} e^{2 c_1} \sqrt {2 x+e^{4 c_1}} \\ y(x)\to \frac {1}{2} e^{2 c_1} \sqrt {2 x+e^{4 c_1}} \\ y(x)\to 0 \\ y(x)\to -\frac {i x}{2} \\ y(x)\to \frac {i x}{2} \\ \end{align*}