83.20.5 problem 5

Internal problem ID [19172]
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 : Monday, March 31, 2025 at 06:51:09 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 45

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

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

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 81

ode=x*(1+D[y[x],x]^2)==1; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 1.392 (sec). Leaf size: 116

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(Derivative(y(x), x)**2 + 1) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = C_{1} + \begin {cases} i \sqrt {x} \sqrt {x - 1} - i \operatorname {acosh}{\left (\sqrt {x} \right )} & \text {for}\: \left |{x}\right | > 1 \\- \frac {x^{\frac {3}{2}}}{\sqrt {1 - x}} + \frac {\sqrt {x}}{\sqrt {1 - x}} + \operatorname {asin}{\left (\sqrt {x} \right )} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = C_{1} - \begin {cases} i \sqrt {x} \sqrt {x - 1} - i \operatorname {acosh}{\left (\sqrt {x} \right )} & \text {for}\: \left |{x}\right | > 1 \\- \frac {x^{\frac {3}{2}}}{\sqrt {1 - x}} + \frac {\sqrt {x}}{\sqrt {1 - x}} + \operatorname {asin}{\left (\sqrt {x} \right )} & \text {otherwise} \end {cases}\right ] \]