Internal problem ID [8055]
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]
Solve \begin {gather*} \boxed {4 y \left (y^{\prime }\right )^{2}+2 x y^{\prime }-y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.078 (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 \relax (x ) = -\frac {i x}{2} \\ y \relax (x ) = \frac {i x}{2} \\ y \relax (x ) = 0 \\ y \relax (x ) = \sqrt {-x c_{1}+c_{1}^{2}} \\ y \relax (x ) = \sqrt {x c_{1}+c_{1}^{2}} \\ y \relax (x ) = -\sqrt {-x c_{1}+c_{1}^{2}} \\ y \relax (x ) = -\sqrt {x c_{1}+c_{1}^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.515 (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*}