44.14.15 problem 2(g)

Internal problem ID [9339]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 2. Problems for Review and Discovery. Drill excercises. Page 105
Problem number : 2(g)
Date solved : Tuesday, September 30, 2025 at 06:16:36 PM
CAS classification : [[_2nd_order, _quadrature]]

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

With initial conditions

\begin{align*} y \left (1\right )&=1 \\ y^{\prime }\left (1\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.294 (sec). Leaf size: 59

ode:=diff(diff(y(x),x),x) = tan(x); 
ic:=[y(1) = 1, D(y)(1) = -1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -\frac {i x^{2}}{2}-x \ln \left (\cos \left (x \right )\right )+x \ln \left (1+{\mathrm e}^{2 i x}\right )-\frac {i \operatorname {polylog}\left (2, -{\mathrm e}^{2 i x}\right )}{2}+x \ln \left (\cos \left (1\right )\right )-x +\frac {i \operatorname {polylog}\left (2, -{\mathrm e}^{2 i}\right )}{2}-\ln \left (1+{\mathrm e}^{2 i}\right )+2+\frac {i}{2} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 32

ode=D[y[x],{x,2}]==Tan[x]; 
ic={y[1]==1,Derivative[1][y][1]==-1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \int _1^x-\log (\cos (K[1]))dK[1]+x (\log (\cos (1))-1)+2-\log (\cos (1)) \end{align*}

✓ Sympy. Time used: 0.583 (sec). Leaf size: 70

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

\[ y{\left (x \right )} = x \left (- \log {\left (\cos {\left (x \right )} \right )} + \frac {- \sin {\left (1 \right )} - \cos {\left (1 \right )} + \log {\left (\cos {\left (1 \right )} \right )} \cos {\left (1 \right )} + \cos {\left (1 \right )} \tan {\left (1 \right )}}{\cos {\left (1 \right )}}\right ) + \frac {\cos {\left (1 \right )} \int \limits ^{1} x \tan {\left (x \right )}\, dx - \cos {\left (1 \right )} \tan {\left (1 \right )} + \sin {\left (1 \right )} + 2 \cos {\left (1 \right )}}{\cos {\left (1 \right )}} - \int x \tan {\left (x \right )}\, dx \]