83.20.5 problem 5

Internal problem ID [19243]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IV. Equations of the first order but not of the first degree. Exercise IV (C) at page 56
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 01:15:32 PM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.018 (sec). Leaf size: 45

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

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

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 81

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

\begin{align*} y(x)\to -2 \arctan \left (\frac {\sqrt {1-x}}{\sqrt {x}+1}\right )+\sqrt {-((x-1) x)}+c_1 \\ y(x)\to 2 \arctan \left (\frac {\sqrt {1-x}}{\sqrt {x}+1}\right )-\sqrt {-((x-1) x)}+c_1 \\ \end{align*}