Internal problem ID [6127]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 16. Nonlinear equations. Miscellaneous Exercises. Page 340
Problem number: 19.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _rational, _Clairaut]
Solve \begin {gather*} \boxed {x^{2} \left (y^{\prime }\right )^{2}-\left (1+2 x y\right ) y^{\prime }+y^{2}+1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.204 (sec). Leaf size: 42
dsolve(x^2*diff(y(x),x)^2-(2*x*y(x)+1)*diff(y(x),x)+y(x)^2+1=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {4 x^{2}-1}{4 x} \\ y \relax (x ) = c_{1} x -\sqrt {c_{1}-1} \\ y \relax (x ) = c_{1} x +\sqrt {c_{1}-1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.563 (sec). Leaf size: 58
DSolve[x^2*y'[x]^2-(2*x*y[x]+1)*y'[x]+y[x]^2+1==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x+\frac {x}{c_1{}^2}+\frac {1}{c_1} \\ y(x)\to x+\frac {x}{4 c_1{}^2}+\frac {1}{2 c_1} \\ y(x)\to x \\ y(x)\to x-\frac {1}{4 x} \\ \end{align*}