Internal problem ID [6887]
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]
\[ \boxed {x {y^{\prime }}^{2}+\left (k -x -y\right ) y^{\prime }+y=0} \]
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 59
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 \left (x \right ) = k +x -2 \sqrt {k x} y \left (x \right ) = k +x +2 \sqrt {k x} y \left (x \right ) = -\frac {\left (c_{1}^{2}-c_{1} \right ) x}{1-c_{1}}-\frac {k c_{1}}{1-c_{1}} \end{align*}
✓ Solution by Mathematica
Time used: 0.014 (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 c_1 \left (x+\frac {k}{-1+c_1}\right ) y(x)\to -2 \sqrt {k} \sqrt {x}+k+x y(x)\to \left (\sqrt {k}+\sqrt {x}\right )^2 \end{align*}