77.32.7 problem 7

Internal problem ID [20691]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Exercise VII (F) at page 113
Problem number : 7
Date solved : Thursday, October 02, 2025 at 06:18:51 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.728 (sec). Leaf size: 45

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

\begin{align*} y &= -i x +c_{1} \\ y &= i x +c_{1} \\ y &= c_{1} +\frac {a^{2} {\mathrm e}^{-\frac {x}{a}}}{4 c_{2}}+c_{2} {\mathrm e}^{\frac {x}{a}} \\ \end{align*}

✓ Mathematica. Time used: 0.385 (sec). Leaf size: 40

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

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

✓ Sympy. Time used: 173.992 (sec). Leaf size: 48

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

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