Internal problem ID [6134]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 16. Nonlinear equations. Miscellaneous Exercises. Page 340
Problem number: 26.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _rational, _dAlembert]
Solve \begin {gather*} \boxed {x \left (y^{\prime }\right )^{2}+\left (k -x -y\right ) y^{\prime }+y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.186 (sec). Leaf size: 44
dsolve(x*diff(y(x),x)^2+(k-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} k}{1-c_{1}} \\ y \relax (x ) = c_{1} \sqrt {x}+k +x \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.013 (sec). Leaf size: 54
DSolve[x*y'[x]^2+(k-x-y[x])*y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to k+\frac {k}{c_1}+x+c_1 x \\ y(x)\to -2 \sqrt {k} \sqrt {x}+k+x \\ y(x)\to \left (\sqrt {k}+\sqrt {x}\right )^2 \\ \end{align*}