54.2.16 problem 19

Internal problem ID [8548]
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
Date solved : Monday, January 27, 2025 at 04:13:56 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

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

✓ Solution by Maple

Time used: 0.032 (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 &= \frac {4 x^{2}-1}{4 x} \\ y &= c_{1} x -\sqrt {c_{1} -1} \\ y &= c_{1} x +\sqrt {c_{1} -1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.953 (sec). Leaf size: 66

DSolve[x^2*D[y[x],x]^2-(2*x*y[x]+1)*D[y[x],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*}