Internal problem ID [6880]
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]
\[ \boxed {x^{2} {y^{\prime }}^{2}-\left (2 y x +1\right ) y^{\prime }+y^{2}=-1} \]
✓ Solution by Maple
Time used: 0.062 (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 \left (x \right ) = \frac {4 x^{2}-1}{4 x} y \left (x \right ) = c_{1} x -\sqrt {c_{1} -1} y \left (x \right ) = c_{1} x +\sqrt {c_{1} -1} \end{align*}
✓ Solution by Mathematica
Time used: 1.527 (sec). Leaf size: 66
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+e^{-2 c_1} x+e^{-c_1} y(x)\to x+\frac {1}{4} e^{-2 c_1} x+\frac {e^{-c_1}}{2} y(x)\to x y(x)\to x-\frac {1}{4 x} \end{align*}