89.24.9 problem 9

Internal problem ID [24846]
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 171
Problem number : 9
Date solved : Thursday, October 02, 2025 at 10:48:36 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 34

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

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

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 30

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

\begin{align*} y(x)&\to -\sin (x) \text {arctanh}(\cos (x))+c_1 \cos (x)+c_2 \sin (x)+\cos (x) \left (-\coth ^{-1}(\sin (x))\right ) \end{align*}

✓ Sympy. Time used: 0.232 (sec). Leaf size: 46

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

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