22.1.2 problem 2

Internal problem ID [4088]
Book : Applied Differential equations, Newby Curle. Van Nostrand Reinhold. 1972
Section : Examples, page 35
Problem number : 2
Date solved : Monday, January 27, 2025 at 08:08:18 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

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

✓ Solution by Maple

Time used: 0.079 (sec). Leaf size: 57

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

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

✓ Solution by Mathematica

Time used: 0.123 (sec). Leaf size: 73

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