Internal problem ID [10202]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IV, differential equations of the first order and higher degree than the first. Article
27. Clairaut equation. Page 56
Problem number: Ex 1.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _rational, _Clairaut]
Solve \begin {gather*} \boxed {\left (y^{\prime } x -y\right )^{2}-\left (y^{\prime }\right )^{2}-1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 46
dsolve((diff(y(x),x)*x-y(x))^2=diff(y(x),x)^2+1,y(x), singsol=all)
\begin{align*} y \relax (x ) = x c_{1}-\sqrt {c_{1}^{2}+1} \\ y \relax (x ) = x c_{1}+\sqrt {c_{1}^{2}+1} \\ y \relax (x ) = \sqrt {x -1}\, \sqrt {1+x}\, c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.131 (sec). Leaf size: 73
DSolve[(y'[x]*x-y[x])^2==(y'[x])^2+1,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 x-\sqrt {1+c_1{}^2} \\ y(x)\to c_1 x+\sqrt {1+c_1{}^2} \\ y(x)\to -\sqrt {1-x^2} \\ y(x)\to \sqrt {1-x^2} \\ \end{align*}