62.16.1 problem Ex 1

Internal problem ID [12892]
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
Date solved : Tuesday, January 28, 2025 at 04:32:12 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

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

✓ Solution by Maple

Time used: 0.138 (sec). Leaf size: 57

dsolve((diff(y(x),x)*x-y(x))^2=diff(y(x),x)^2+1,y(x), singsol=all)
 

\begin{align*} y &= \sqrt {-x^{2}+1} \\ y &= -\sqrt {-x^{2}+1} \\ y &= c_{1} x -\sqrt {c_{1}^{2}+1} \\ y &= c_{1} x +\sqrt {c_{1}^{2}+1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.138 (sec). Leaf size: 73

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