77.20.5 problem 5

Internal problem ID [20528]
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 : Thursday, October 02, 2025 at 06:04:32 PM
CAS classification : [_quadrature]

\begin{align*} \left (1+{y^{\prime }}^{2}\right ) x&=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: 0.006 (sec). Leaf size: 71

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

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

✓ Sympy. Time used: 0.888 (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)
 

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