53.3.10 problem 12

Internal problem ID [8472]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 99. Clairaut equation. EXERCISES Page 320
Problem number : 12
Date solved : Monday, January 27, 2025 at 04:06:35 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, _dAlembert]

\begin{align*} x {y^{\prime }}^{2}+\left (x -y\right ) y^{\prime }+1-y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.052 (sec). Leaf size: 44

dsolve(x*diff(y(x),x)^2+(x-y(x))*diff(y(x),x)+1-y(x)=0,y(x), singsol=all)
 

\begin{align*} y &= -x -2 \sqrt {x} \\ y &= -x +2 \sqrt {x} \\ y &= \frac {c_{1}^{2} x +c_{1} x +1}{c_{1} +1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.019 (sec). Leaf size: 46

DSolve[x*(D[y[x],x])^2+(x-y[x])*D[y[x],x]+1-y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to c_1 x+\frac {1}{1+c_1} \\ y(x)\to -x-2 \sqrt {x} \\ y(x)\to 2 \sqrt {x}-x \\ \end{align*}