83.46.2 problem Ex 2 page 69

Internal problem ID [19567]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Book Solved Excercises. Chapter V. Singular solutions
Problem number : Ex 2 page 69
Date solved : Tuesday, January 28, 2025 at 01:57:05 PM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.046 (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: 0.211 (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*}