44.11.12 problem 2(f)

Internal problem ID [9282]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 2. Second-Order Linear Equations. Section 2.3. THE METHOD OF VARIATION OF PARAMETERS. Page 71
Problem number : 2(f)
Date solved : Tuesday, September 30, 2025 at 06:16:04 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 23

ode:=diff(diff(y(x),x),x)+y(x) = sec(x)*tan(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \ln \left (\sec \left (x \right )\right ) \sin \left (x \right )+\left (c_2 -1\right ) \sin \left (x \right )+\cos \left (x \right ) \left (x +c_1 \right ) \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 29

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

\begin{align*} y(x)&\to \cos (x) \arctan (\tan (x))+c_1 \cos (x)+\sin (x) (-\log (\cos (x))-1+c_2) \end{align*}

✓ Sympy. Time used: 0.153 (sec). Leaf size: 19

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

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