Internal problem ID [6051]
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’s equation. EXERCISES Page
320
Problem number: 12.
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 (x -y\right ) y^{\prime }+1-y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.328 (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 \relax (x ) = \frac {\left (-c_{1}^{2}-c_{1}\right ) x}{-c_{1}-1}-\frac {1}{-c_{1}-1} \\ y \relax (x ) = -x +\sqrt {x}\, c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.017 (sec). Leaf size: 46
DSolve[x*(y'[x])^2+(x-y[x])*y'[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*}