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 : Monday, January 27, 2025 at 07:29:58 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*}

✓ Solution by Maple

Time used: 0.022 (sec). Leaf size: 14

dsolve([diff(y(x),x$2)=sec(x)*tan(x),y(0) = 1/4*Pi, D(y)(0) = 1],y(x), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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