83.18.6 problem 6

Internal problem ID [19137]
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 (A) at page 53
Problem number : 6
Date solved : Thursday, March 13, 2025 at 01:45:06 PM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{2}-2 y^{\prime } \cosh \left (x \right )+1&=0 \end{align*}

✓ Maple. Time used: 0.043 (sec). Leaf size: 19

ode:=diff(y(x),x)^2-2*diff(y(x),x)*cosh(x)+1 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y \left (x \right ) &= -{\mathrm e}^{-x}+c_{1} \\ y \left (x \right ) &= {\mathrm e}^{x}+c_{1} \\ \end{align*}

✓ Mathematica. Time used: 0.157 (sec). Leaf size: 109

ode=D[y[x],x]^2-D[y[x],x]*Cosh[x]+1==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {1}{2} \left (\sinh (x)-\frac {i \sqrt {3} \sqrt {7-\cosh (2 x)} E\left (i x\left |-\frac {1}{3}\right .\right )}{\sqrt {\cosh (2 x)-7}}\right )+c_1 \\ y(x)\to \frac {1}{2} \left (\sinh (x)+\frac {i \sqrt {3} \sqrt {7-\cosh (2 x)} E\left (i x\left |-\frac {1}{3}\right .\right )}{\sqrt {\cosh (2 x)-7}}\right )+c_1 \\ \end{align*}

✓ Sympy. Time used: 0.470 (sec). Leaf size: 44

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

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