89.25.5 problem 6

Internal problem ID [24863]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 11. Variation of parameters and other methods. Exercises at page 177
Problem number : 6
Date solved : Thursday, October 02, 2025 at 10:48:48 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 22

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

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

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 22

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

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

✓ Sympy. Time used: 0.131 (sec). Leaf size: 19

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

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