15.18.29 problem 29

Internal problem ID [3272]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 35, page 157
Problem number : 29
Date solved : Sunday, March 30, 2025 at 01:26:00 AM
CAS classification : [[_2nd_order, _quadrature]]

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

With initial conditions

\begin{align*} y \left (0\right )&=\frac {\pi }{4}\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.035 (sec). Leaf size: 14

ode:=diff(diff(y(x),x),x) = sec(x)*tan(x); 
ic:=y(0) = 1/4*Pi, D(y)(0) = 1; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \ln \left (\sec \left (x \right )+\tan \left (x \right )\right )+\frac {\pi }{4} \]

✓ Mathematica. Time used: 0.048 (sec). Leaf size: 20

ode=D[y[x],{x,2}]==Sec[x]*Tan[x]; 
ic={y[0]==Pi/4,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{4} \left (8 \text {arctanh}\left (\tan \left (\frac {x}{2}\right )\right )+\pi \right ) \]

✓ Sympy. Time used: 0.714 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) - tan(x)/cos(x),0) 
ics = {y(0): pi/4, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {x}{\cos {\left (x \right )}} - \int \frac {x \tan {\left (x \right )}}{\cos {\left (x \right )}}\, dx + \int \limits ^{0} \frac {x \tan {\left (x \right )}}{\cos {\left (x \right )}}\, dx + \frac {\pi }{4} \]