82.20.1 problem Ex. 1

Internal problem ID [18853]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter IV. Singular solutions. problems at page 45
Problem number : Ex. 1
Date solved : Tuesday, January 28, 2025 at 12:30:50 PM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.034 (sec). Leaf size: 54

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

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

✓ Solution by Mathematica

Time used: 60.034 (sec). Leaf size: 55

DSolve[y[x]^2*(1+D[y[x],x])^2==r^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -r \left (1+W\left (-\frac {e^{-\frac {r-x+c_1}{r}}}{r}\right )\right ) \\ y(x)\to r \left (1+W\left (\frac {e^{-\frac {r+x+c_1}{r}}}{r}\right )\right ) \\ \end{align*}