82.18.23 problem Ex. 25

Internal problem ID [18774]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter III. Equations of the first order but not of the first degree. Examples on chapter III. page 38
Problem number : Ex. 25
Date solved : Monday, March 31, 2025 at 06:10:28 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.362 (sec). Leaf size: 51

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

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

✓ Mathematica. Time used: 0.438 (sec). Leaf size: 124

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

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

✓ Sympy. Time used: 25.763 (sec). Leaf size: 68

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

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