Internal problem ID [8005]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 425.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _rational, _dAlembert]
Solve \begin {gather*} \boxed {\left (x +1\right ) \left (y^{\prime }\right )^{2}-\left (x +y\right ) y^{\prime }+y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.256 (sec). Leaf size: 46
dsolve((x+1)*diff(y(x),x)^2-(x+y(x))*diff(y(x),x)+y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (-c_{1}^{2}+c_{1}\right ) x}{1-c_{1}}-\frac {c_{1}^{2}}{1-c_{1}} \\ y \relax (x ) = \sqrt {x +1}\, c_{1}+x +2 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.012 (sec). Leaf size: 49
DSolve[y[x] - (x + y[x])*y'[x] + (1 + x)*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \left (x+1+\frac {1}{-1+c_1}\right ) \\ y(x)\to x-2 \sqrt {x+1}+2 \\ y(x)\to x+2 \sqrt {x+1}+2 \\ \end{align*}