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: 0.063 (sec). Leaf size: 45
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 {c_{1} \left (c_{1} x +k -x \right )}{c_{1} -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*}