30.1.8 problem example page 46

Internal problem ID [5696]
Book : Differential and integral calculus, vol II By N. Piskunov. 1974
Section : Chapter 1
Problem number : example page 46
Date solved : Monday, January 27, 2025 at 01:07:32 PM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.062 (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 {R^{2}-c_{1}^{2}+2 c_{1} x -x^{2}} \\ y \left (x \right ) &= -\sqrt {\left (R +c_{1} -x \right ) \left (R -c_{1} +x \right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.246 (sec). Leaf size: 101

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

\begin{align*} y(x)\to -\sqrt {R^2-(x+c_1){}^2} \\ y(x)\to \sqrt {R^2-(x+c_1){}^2} \\ y(x)\to -\sqrt {R^2-(x-c_1){}^2} \\ y(x)\to \sqrt {R^2-(x-c_1){}^2} \\ y(x)\to -R \\ y(x)\to R \\ \end{align*}