54.2.23 problem 26

Internal problem ID [8555]
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
Date solved : Monday, January 27, 2025 at 04:14:11 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, _dAlembert]

\begin{align*} x {y^{\prime }}^{2}+\left (k -x -y\right ) y^{\prime }+y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.064 (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 &= k +x -2 \sqrt {k x} \\ y &= k +x +2 \sqrt {k x} \\ y &= \frac {c_{1} \left (c_{1} x +k -x \right )}{c_{1} -1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.017 (sec). Leaf size: 54

DSolve[x*D[y[x],x]^2+(k-x-y[x])*D[y[x],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*}