76.46.2 problem Ex. 2

Internal problem ID [20258]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VIII. Exact differential equations, and equations of particular forms. Integration in series. problems at page 101
Problem number : Ex. 2
Date solved : Thursday, October 02, 2025 at 05:38:00 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.537 (sec). Leaf size: 32

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

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

✓ Mathematica. Time used: 0.319 (sec). Leaf size: 19

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

\begin{align*} y(x)&\to a \cosh \left (\frac {x}{a}+c_1\right )+c_2 \end{align*}

✓ Sympy. Time used: 176.117 (sec). Leaf size: 48

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

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