83.48.12 problem Ex 12 page 107

Internal problem ID [19533]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Book Solved Excercises. Chapter VII. Exact differential equations.
Problem number : Ex 12 page 107
Date solved : Monday, March 31, 2025 at 07:32:10 PM
CAS classification : [[_2nd_order, _quadrature]]

\begin{align*} y^{\prime \prime }&=\sec \left (x \right )^{2} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 14

ode:=diff(diff(y(x),x),x) = sec(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\ln \left (\cos \left (x \right )\right )+c_1 x +c_2 \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 17

ode=D[y[x],{x,2}]==Sec[x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_2 x-\log (\cos (x))+c_1 \]

✓ Sympy. Time used: 0.797 (sec). Leaf size: 129

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

\[ y{\left (x \right )} = C_{1} + C_{2} x - \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )}}{2} + \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )}}{2 \cos {\left (x \right )}} - \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}}{2} + \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}}{2 \cos {\left (x \right )}} + \frac {\log {\left (\frac {2}{\cos {\left (x \right )} + 1} \right )} \tan ^{2}{\left (\frac {x}{2} \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} - 1} - \frac {\log {\left (\frac {2}{\cos {\left (x \right )} + 1} \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} - 1} + \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} - 1} + \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} - 1} \]