83.22.2 problem 2

Internal problem ID [19185]
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 (E) at page 63
Problem number : 2
Date solved : Monday, March 31, 2025 at 06:51:35 PM
CAS classification : [_quadrature]

\begin{align*} x +\frac {y^{\prime }}{\sqrt {1+{y^{\prime }}^{2}}}&=a \end{align*}

✓ Maple. Time used: 0.047 (sec). Leaf size: 40

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

\[ y = -\left (a -x +1\right ) \left (a -x -1\right ) \sqrt {-\frac {1}{\left (a -x +1\right ) \left (a -x -1\right )}}+c_1 \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 55

ode=x+D[y[x],x]/Sqrt[1+D[y[x],x]^2]==a; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to c_1-i \sqrt {a^2-2 a x+x^2-1} \\ y(x)\to i \sqrt {a^2-2 a x+x^2-1}+c_1 \\ \end{align*}

✓ Sympy. Time used: 0.901 (sec). Leaf size: 194

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

\[ \left [ y{\left (x \right )} = C_{1} + a^{2} \sqrt {- \frac {1}{a^{2} - 2 a x + x^{2} - 1}} - 2 a x \sqrt {- \frac {1}{a^{2} - 2 a x + x^{2} - 1}} + x^{2} \sqrt {- \frac {1}{a^{2} - 2 a x + x^{2} - 1}} - \sqrt {- \frac {1}{a^{2} - 2 a x + x^{2} - 1}}, \ y{\left (x \right )} = C_{1} - a^{2} \sqrt {- \frac {1}{a^{2} - 2 a x + x^{2} - 1}} + 2 a x \sqrt {- \frac {1}{a^{2} - 2 a x + x^{2} - 1}} - x^{2} \sqrt {- \frac {1}{a^{2} - 2 a x + x^{2} - 1}} + \sqrt {- \frac {1}{a^{2} - 2 a x + x^{2} - 1}}\right ] \]